di erential equations, make up test 1. - uni-miskolc.hu
TRANSCRIPT
Differential equations, Make up test 1.
May 17, 2021
1
1.
Feladat. 1. Let y = (ay − b)(c − y), where a = 2, b = 5, c = 3 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 0 , B: 1 , C: 2 , D: 3 , E: 4
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 2, b = 5, c = 2 . How much is u(2, 3)?
Valaszok. A: 0.0810619 , B: 0.0910326 , C: 0.10223 , D: 0.114804 , E: 0.128925
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.04, b = 8, c = 7 . How much is x13 ?
Valaszok. A: 144.67 , B: 162.465 , C: 182.448 , D: 204.889 , E: 230.09
Feladat. 4. Let y = tayb, y(2) = c, where a = 2, b = 1, c = 1 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 0.654539 , B: 0.735047 , C: 0.825458 , D: 0.926989 , E: 1.04101
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 7, c = 8 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 22 , B: 23 , C: 24 , D: 25 , E: 26
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 70 10
), ~y0 =
(2318
).
How much is y1(0.1) ?
Valaszok. A: 37.7677 , B: 42.4132 , C: 47.63 , D: 53.4885 , E: 60.0676
Feladat. 7. Let
A =
(8 32 6
).
How much is det(e0.5A
)?
Valaszok. A: 1096.63 , B: 1231.52 , C: 1383. , D: 1553.1 , E: 1744.14
Feladat. 8. Let a = −3, b = −1, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00552456 , B: 0.00620408 , C: 0.00696718 , D: 0.00782414 , E: 0.00878651
Feladat. 9. Let a = 25, t0 = 7. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0762569 , B: −0.0856365 , C: −0.0961698 , D: −0.107999 , E: −0.121282
Feladat. 10. Let a = 3, b = 5, t0 = 2.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.173906 , B: 0.195296 , C: 0.219318 , D: 0.246294 , E: 0.276588
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 6, b = 4, c = 9, d = 4 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 15340 , B: 157
40 , C: 15940 , D: 161
40 , E: 16340
Feladat. 12. Let a = 3, b = 6, c = 7, d = 7, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −0.796784 , B: −0.894788 , C: −1.00485 , D: −1.12844 , E: −1.26724
2
2.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 2, c = 5 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 12 , B: 13 , C: 14 , D: 15 , E: 16
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 2, c = 4 . How much is u(2, 3)?
Valaszok. A: 0.225611 , B: 0.253361 , C: 0.284525 , D: 0.319521 , E: 0.358822
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.04, b = 4, c = 7 . How much is x13 ?
Valaszok. A: 78.1629 , B: 87.7769 , C: 98.5735 , D: 110.698 , E: 124.314
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 2, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.31 , B: 1.47113 , C: 1.65208 , D: 1.85528 , E: 2.08348
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 1, b = 5, c = 8 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 50 , B: 51 , C: 52 , D: 53 , E: 54
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(3 60 10
), ~y0 =
(2121
).
How much is y1(0.1) ?
Valaszok. A: 47.176 , B: 52.9786 , C: 59.495 , D: 66.8129 , E: 75.0309
Feladat. 7. Let
A =
(1 57 4
).
How much is det(e0.5A
)?
Valaszok. A: 9.65999 , B: 10.8482 , C: 12.1825 , D: 13.6809 , E: 15.3637
Feladat. 8. Let a = −1, b = 0, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.043853 , B: 0.0492469 , C: 0.0553043 , D: 0.0621067 , E: 0.0697459
Feladat. 9. Let a = 16, t0 = −2. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0)?
Valaszok. A: −0.085 , B: −0.0425 , C: 0. , D: 0.0425 , E: 0.085
Feladat. 10. Let a = 6, b = 6, t0 = 4.75. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.131013 , B: 0.147128 , C: 0.165225 , D: 0.185547 , E: 0.20837
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 3, b = 7, c = 9, d = 5 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 9934 , B: 101
34 , C: 10334 , D: 105
34 , E: 10734
Feladat. 12. Let a = 5, b = 3, c = 2, d = 4, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 0.71225 , B: 0.799856 , C: 0.898239 , D: 1.00872 , E: 1.13279
3
3.
Feladat. 1. Let y = (ay − b)(c − y), where a = 5, b = 4, c = 4 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 13 , B: 14 , C: 15 , D: 16 , E: 17
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 3, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.105138 , B: 0.11807 , C: 0.132593 , D: 0.148902 , E: 0.167217
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.01, b = 1, c = 8 . How much is x13 ?
Valaszok. A: 20.4043 , B: 22.9141 , C: 25.7325 , D: 28.8976 , E: 32.452
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 2, c = 3 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.25358 , B: 2.53077 , C: 2.84206 , D: 3.19163 , E: 3.5842
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 7, c = 11 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 62 , B: 63 , C: 64 , D: 65 , E: 66
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 30 7
), ~y0 =
(1015
).
How much is y1(0.1) ?
Valaszok. A: 17.2263 , B: 19.3452 , C: 21.7246 , D: 24.3968 , E: 27.3976
Feladat. 7. Let
A =
(9 35 2
).
How much is det(e0.5A
)?
Valaszok. A: 244.692 , B: 274.789 , C: 308.588 , D: 346.544 , E: 389.169
Feladat. 8. Let a = −2, b = 2, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0012767 , B: 0.00143374 , C: 0.00161009 , D: 0.00180813 , E: 0.00203053
Feladat. 9. Let a = 16, t0 = 7. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: 0.0677265 , B: 0.0760569 , C: 0.0854119 , D: 0.0959175 , E: 0.107715
Feladat. 10. Let a = 5, b = 2, t0 = −1.75. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: −0.17 , B: −0.1275 , C: −0.085 , D: −0.0425 , E: 0.
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 6, b = 7, c = 8, d = 3 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 13746 , B: 139
46 , C: 14146 , D: 143
46 , E: 14546
Feladat. 12. Let a = 3, b = 6, c = 3, d = 7, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −0.796432 , B: −0.894394 , C: −1.0044 , D: −1.12795 , E: −1.26668
4
4.
Feladat. 1. Let y = (ay − b)(c − y), where a = 5, b = 4, c = 3 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 10 , B: 11 , C: 12 , D: 13 , E: 14
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 5, b = 5, c = 3 . How much is u(2, 3)?
Valaszok. A: 0.157203 , B: 0.176539 , C: 0.198253 , D: 0.222638 , E: 0.250023
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.01, b = 7, c = 1 . How much is x13 ?
Valaszok. A: 97.8034 , B: 109.833 , C: 123.343 , D: 138.514 , E: 155.551
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 1, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.5306 , B: 1.71886 , C: 1.93028 , D: 2.1677 , E: 2.43433
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 7, c = 11 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 63 , B: 64 , C: 65 , D: 66 , E: 67
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(3 50 9
), ~y0 =
(76
).
How much is y1(0.1) ?
Valaszok. A: 13.3551 , B: 14.9977 , C: 16.8425 , D: 18.9141 , E: 21.2405
Feladat. 7. Let
A =
(9 36 1
).
How much is det(e0.5A
)?
Valaszok. A: 117.683 , B: 132.158 , C: 148.413 , D: 166.668 , E: 187.168
Feladat. 8. Let a = 0, b = 2, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00491946 , B: 0.00552456 , C: 0.00620408 , D: 0.00696718 , E: 0.00782414
Feladat. 9. Let a = 9, t0 = 1. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: 0.0418878 , B: 0.04704 , C: 0.0528259 , D: 0.0593235 , E: 0.0666203
Feladat. 10. Let a = 2, b = 3, t0 = −0.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: −0.17 , B: −0.1275 , C: −0.085 , D: −0.0425 , E: 0.
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 8, b = 9, c = 3, d = 7 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 8158 , B: 83
58 , C: 8558 , D: 89
58 , E: 9158
Feladat. 12. Let a = 2, b = 3, c = 3, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −0.478999 , B: −0.537916 , C: −0.60408 , D: −0.678382 , E: −0.761823
5
5.
Feladat. 1. Let y = (ay − b)(c − y), where a = 5, b = 2, c = 5 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 23 , B: 24 , C: 25 , D: 26 , E: 27
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 2, b = 3, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.113123 , B: 0.127037 , C: 0.142663 , D: 0.16021 , E: 0.179916
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.08, b = 8, c = 9 . How much is x13 ?
Valaszok. A: 138.704 , B: 155.764 , C: 174.923 , D: 196.439 , E: 220.601
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 3, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.72963 , B: 1.94237 , C: 2.18128 , D: 2.44958 , E: 2.75088
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 2, b = 4, c = 7 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 26 , B: 27 , C: 28 , D: 29 , E: 30
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 60 8
), ~y0 =
(42
).
How much is y1(0.1) ?
Valaszok. A: 7.27377 , B: 8.16845 , C: 9.17317 , D: 10.3015 , E: 11.5685
Feladat. 7. Let
A =
(7 62 1
).
How much is det(e0.5A
)?
Valaszok. A: 48.6181 , B: 54.5982 , C: 61.3137 , D: 68.8553 , E: 77.3245
Feladat. 8. Let a = −2, b = 2, c = 3, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00444728 , B: 0.0049943 , C: 0.00560859 , D: 0.00629845 , E: 0.00707316
Feladat. 9. Let a = 9, t0 = 1. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: 0.0332145 , B: 0.0372999 , C: 0.0418878 , D: 0.04704 , E: 0.0528259
Feladat. 10. Let a = 5, b = 4, t0 = −0.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: −0.0425 , B: 0. , C: 0.0425 , D: 0.085 , E: 0.1275
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 2, b = 8, c = 2, d = 3 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 2732 , B: 29
32 , C: 3132 , D: 33
32 , E: 3532
Feladat. 12. Let a = 4, b = 6, c = 4, d = 7, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −1.19252 , B: −1.33921 , C: −1.50393 , D: −1.68891 , E: −1.89665
6
6.
Feladat. 1. Let y = (ay − b)(c − y), where a = 5, b = 4, c = 4 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 13 , B: 14 , C: 15 , D: 16 , E: 17
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 5, b = 5, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.104799 , B: 0.117689 , C: 0.132165 , D: 0.148421 , E: 0.166677
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.01, b = 1, c = 5 . How much is x13 ?
Valaszok. A: 15.4622 , B: 17.364 , C: 19.4998 , D: 21.8983 , E: 24.5918
Feladat. 4. Let y = tayb, y(2) = c, where a = 2, b = 2, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.36707 , B: 1.53522 , C: 1.72405 , D: 1.93611 , E: 2.17425
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 4, b = 8, c = 11 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 50 , B: 51 , C: 52 , D: 53 , E: 54
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(3 50 7
), ~y0 =
(138
).
How much is y1(0.1) ?
Valaszok. A: 19.1789 , B: 21.5379 , C: 24.1871 , D: 27.1621 , E: 30.5031
Feladat. 7. Let
A =
(8 58 8
).
How much is det(e0.5A
)?
Valaszok. A: 2654.46 , B: 2980.96 , C: 3347.62 , D: 3759.37 , E: 4221.78
Feladat. 8. Let a = −2, b = 1, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0340768 , B: 0.0382683 , C: 0.0429753 , D: 0.0482613 , E: 0.0541974
Feladat. 9. Let a = 25, t0 = 10. Find the retarded solution of the G′′(t) +aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0467276 , B: −0.0524751 , C: −0.0589295 , D: −0.0661779 , E: −0.0743178
Feladat. 10. Let a = 2, b = 3, t0 = −1.75. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: −0.1275 , B: −0.085 , C: −0.0425 , D: 0. , E: 0.0425
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 9, b = 9, c = 4, d = 2 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 9762 , B: 99
62 , C: 10162 , D: 103
62 , E: 10562
Feladat. 12. Let a = 7, b = 2, c = 4, d = 4, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 1.45584 , B: 1.63491 , C: 1.836 , D: 2.06183 , E: 2.31543
7
7.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 3, c = 2 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 0 , B: 1 , C: 2 , D: 3 , E: 4
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 4, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.131936 , B: 0.148165 , C: 0.166389 , D: 0.186855 , E: 0.209838
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.06, b = 8, c = 5 . How much is x13 ?
Valaszok. A: 128.236 , B: 144.009 , C: 161.722 , D: 181.614 , E: 203.952
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 1, c = 3 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.42699 , B: 2.72551 , C: 3.06075 , D: 3.43723 , E: 3.86
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 4, c = 8 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 23 , B: 24 , C: 25 , D: 26 , E: 27
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 60 7
), ~y0 =
(85
).
How much is y1(0.1) ?
Valaszok. A: 9.13286 , B: 10.2562 , C: 11.5177 , D: 12.9344 , E: 14.5253
Feladat. 7. Let
A =
(8 67 4
).
How much is det(e0.5A
)?
Valaszok. A: 284.857 , B: 319.895 , C: 359.242 , D: 403.429 , E: 453.051
Feladat. 8. Let a = −2, b = 3, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0291174 , B: 0.0326988 , C: 0.0367208 , D: 0.0412374 , E: 0.0463096
Feladat. 9. Let a = 36, t0 = −1. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0)?
Valaszok. A: −0.085 , B: −0.0425 , C: 0. , D: 0.0425 , E: 0.085
Feladat. 10. Let a = 3, b = 2, t0 = 0.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.0482626 , B: 0.0541989 , C: 0.0608654 , D: 0.0683519 , E: 0.0767591
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 5, b = 6, c = 4, d = 7 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 7740 , B: 79
40 , C: 8140 , D: 83
40 , E: 8740
Feladat. 12. Let a = 6, b = 4, c = 2, d = 5, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −3.67146 , B: −4.12305 , C: −4.63018 , D: −5.19969 , E: −5.83925
8
8.
Feladat. 1. Let y = (ay − b)(c − y), where a = 5, b = 5, c = 5 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 18 , B: 19 , C: 20 , D: 21 , E: 22
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 2, b = 3, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.113123 , B: 0.127037 , C: 0.142663 , D: 0.16021 , E: 0.179916
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.05, b = 4, c = 5 . How much is x13 ?
Valaszok. A: 63.6574 , B: 71.4872 , C: 80.2802 , D: 90.1546 , E: 101.244
Feladat. 4. Let y = tayb, y(2) = c, where a = 2, b = 1, c = 3 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.20514 , B: 2.47637 , C: 2.78097 , D: 3.12303 , E: 3.50716
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 1, b = 5, c = 6 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 26 , B: 27 , C: 28 , D: 29 , E: 30
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 80 10
), ~y0 =
(159
).
How much is y1(0.1) ?
Valaszok. A: 26.1923 , B: 29.414 , C: 33.0319 , D: 37.0949 , E: 41.6575
Feladat. 7. Let
A =
(3 82 1
).
How much is det(e0.5A
)?
Valaszok. A: 6.57975 , B: 7.38906 , C: 8.29791 , D: 9.31855 , E: 10.4647
Feladat. 8. Let a = −2, b = 1, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00563286 , B: 0.00632571 , C: 0.00710377 , D: 0.00797753 , E: 0.00895877
Feladat. 9. Let a = 36, t0 = 7. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.136023 , B: −0.152754 , C: −0.171542 , D: −0.192642 , E: −0.216337
Feladat. 10. Let a = 5, b = 6, t0 = 0.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.0384657 , B: 0.043197 , C: 0.0485102 , D: 0.0544769 , E: 0.0611776
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 6, b = 4, c = 5, d = 8 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 9740 , B: 99
40 , C: 10140 , D: 103
40 , E: 10740
Feladat. 12. Let a = 6, b = 3, c = 3, d = 3, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 1.28479 , B: 1.44282 , C: 1.62028 , D: 1.81958 , E: 2.04339
9
9.
Feladat. 1. Let y = (ay − b)(c − y), where a = 2, b = 4, c = 4 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 2 , B: 3 , C: 4 , D: 5 , E: 6
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 4, b = 3, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.142599 , B: 0.160139 , C: 0.179836 , D: 0.201956 , E: 0.226797
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.08, b = 6, c = 6 . How much is x13 ?
Valaszok. A: 102.588 , B: 115.206 , C: 129.376 , D: 145.29 , E: 163.16
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 2, c = 1 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 0.720535 , B: 0.809161 , C: 0.908687 , D: 1.02046 , E: 1.14597
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 7, c = 8 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 24 , B: 25 , C: 26 , D: 27 , E: 28
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 60 7
), ~y0 =
(1410
).
How much is y1(0.1) ?
Valaszok. A: 16.7298 , B: 18.7876 , C: 21.0984 , D: 23.6935 , E: 26.6078
Feladat. 7. Let
A =
(9 73 6
).
How much is det(e0.5A
)?
Valaszok. A: 1610.01 , B: 1808.04 , C: 2030.43 , D: 2280.17 , E: 2560.64
Feladat. 8. Let a = −3, b = 0, c = 3, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0109859 , B: 0.0123371 , C: 0.0138546 , D: 0.0155587 , E: 0.0174725
Feladat. 9. Let a = 16, t0 = 8. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: 0.109312 , B: 0.122757 , C: 0.137857 , D: 0.154813 , E: 0.173855
Feladat. 10. Let a = 4, b = 2, t0 = 4.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.27683 , B: 0.31088 , C: 0.349119 , D: 0.39206 , E: 0.440283
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 5, b = 5, c = 4, d = 4 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 6338 , B: 65
38 , C: 6738 , D: 69
38 , E: 7138
Feladat. 12. Let a = 5, b = 7, c = 7, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 0.935959 , B: 1.05108 , C: 1.18037 , D: 1.32555 , E: 1.48859
10
10.
Feladat. 1. Let y = (ay − b)(c − y), where a = 5, b = 4, c = 2 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 5 , B: 6 , C: 7 , D: 8 , E: 9
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 4, b = 5, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.120895 , B: 0.135765 , C: 0.152464 , D: 0.171217 , E: 0.192276
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.01, b = 4, c = 4 . How much is x13 ?
Valaszok. A: 47.4097 , B: 53.241 , C: 59.7897 , D: 67.1438 , E: 75.4025
Feladat. 4. Let y = tayb, y(2) = c, where a = 2, b = 1, c = 1 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 0.654539 , B: 0.735047 , C: 0.825458 , D: 0.926989 , E: 1.04101
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 5, c = 8 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 25 , B: 26 , C: 27 , D: 28 , E: 29
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 70 10
), ~y0 =
(1612
).
How much is y1(0.1) ?
Valaszok. A: 36.5446 , B: 41.0396 , C: 46.0875 , D: 51.7562 , E: 58.1222
Feladat. 7. Let
A =
(8 91 3
).
How much is det(e0.5A
)?
Valaszok. A: 172.775 , B: 194.026 , C: 217.891 , D: 244.692 , E: 274.789
Feladat. 8. Let a = 2, b = 3, c = 3, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0158765 , B: 0.0178293 , C: 0.0200223 , D: 0.0224851 , E: 0.0252507
Feladat. 9. Let a = 4, t0 = 8. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0905102 , B: −0.101643 , C: −0.114145 , D: −0.128185 , E: −0.143952
Feladat. 10. Let a = 5, b = 3, t0 = 3.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.201878 , B: 0.226708 , C: 0.254594 , D: 0.285909 , E: 0.321075
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 3, b = 5, c = 2, d = 7 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 2330 , B: 29
30 , C: 3130 , D: 37
30 , E: 4130
Feladat. 12. Let a = 2, b = 2, c = 2, d = 4, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 0.255508 , B: 0.286935 , C: 0.322228 , D: 0.361862 , E: 0.406371
11
11.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 3, c = 3 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 2 , B: 3 , C: 4 , D: 5 , E: 6
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 5, c = 3 . How much is u(2, 3)?
Valaszok. A: 0.123026 , B: 0.138158 , C: 0.155152 , D: 0.174236 , E: 0.195667
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.08, b = 2, c = 1 . How much is x13 ?
Valaszok. A: 32.2756 , B: 36.2455 , C: 40.7037 , D: 45.7102 , E: 51.3326
Feladat. 4. Let y = tayb, y(2) = c, where a = 2, b = 3, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.70327 , B: 1.91277 , C: 2.14804 , D: 2.41225 , E: 2.70895
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 4, b = 6, c = 9 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 25 , B: 26 , C: 27 , D: 28 , E: 29
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 50 6
), ~y0 =
(1612
).
How much is y1(0.1) ?
Valaszok. A: 20.1611 , B: 22.641 , C: 25.4258 , D: 28.5532 , E: 32.0652
Feladat. 7. Let
A =
(8 69 6
).
How much is det(e0.5A
)?
Valaszok. A: 774.323 , B: 869.565 , C: 976.521 , D: 1096.63 , E: 1231.52
Feladat. 8. Let a = −2, b = 3, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0412374 , B: 0.0463096 , C: 0.0520057 , D: 0.0584024 , E: 0.0655859
Feladat. 9. Let a = 25, t0 = −2. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0)?
Valaszok. A: −0.17 , B: −0.1275 , C: −0.085 , D: −0.0425 , E: 0.
Feladat. 10. Let a = 4, b = 2, t0 = 4.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.27683 , B: 0.31088 , C: 0.349119 , D: 0.39206 , E: 0.440283
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 7, b = 2, c = 5, d = 6 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 10940 , B: 111
40 , C: 11340 , D: 117
40 , E: 11940
Feladat. 12. Let a = 2, b = 2, c = 7, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −1.58386 , B: −1.77867 , C: −1.99745 , D: −2.24314 , E: −2.51904
12
12.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 4, c = 2 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: −1 , B: 0 , C: 1 , D: 2 , E: 3
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 5, b = 2, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.193803 , B: 0.217641 , C: 0.244411 , D: 0.274473 , E: 0.308233
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.02, b = 7, c = 9 . How much is x13 ?
Valaszok. A: 114.405 , B: 128.477 , C: 144.279 , D: 162.026 , E: 181.955
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 2, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.37854 , B: 2.6711 , C: 2.99965 , D: 3.3686 , E: 3.78294
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 1, b = 5, c = 7 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 33 , B: 34 , C: 35 , D: 36 , E: 37
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 60 8
), ~y0 =
(84
).
How much is y1(0.1) ?
Valaszok. A: 10.2719 , B: 11.5353 , C: 12.9542 , D: 14.5475 , E: 16.3369
Feladat. 7. Let
A =
(2 34 9
).
How much is det(e0.5A
)?
Valaszok. A: 194.026 , B: 217.891 , C: 244.692 , D: 274.789 , E: 308.588
Feladat. 8. Let a = −3, b = 2, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0068165 , B: 0.00765493 , C: 0.00859649 , D: 0.00965385 , E: 0.0108413
Feladat. 9. Let a = 4, t0 = 3. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.139708 , B: −0.156892 , C: −0.17619 , D: −0.197861 , E: −0.222198
Feladat. 10. Let a = 2, b = 6, t0 = 2.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.117544 , B: 0.132002 , C: 0.148238 , D: 0.166471 , E: 0.186947
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 8, b = 5, c = 4, d = 5 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 8750 , B: 89
50 , C: 9150 , D: 93
50 , E: 9750
Feladat. 12. Let a = 4, b = 3, c = 2, d = 6, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 0.916998 , B: 1.02979 , C: 1.15645 , D: 1.2987 , E: 1.45844
13
13.
Feladat. 1. Let y = (ay − b)(c − y), where a = 4, b = 3, c = 3 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 5 , B: 6 , C: 7 , D: 8 , E: 9
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 2, b = 3, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.127037 , B: 0.142663 , C: 0.16021 , D: 0.179916 , E: 0.202046
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.09, b = 5, c = 6 . How much is x13 ?
Valaszok. A: 83.726 , B: 94.0243 , C: 105.589 , D: 118.577 , E: 133.162
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 1, c = 3 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.72551 , B: 3.06075 , C: 3.43723 , D: 3.86 , E: 4.33478
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 2, b = 6, c = 9 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 48 , B: 49 , C: 50 , D: 51 , E: 52
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 60 7
), ~y0 =
(1510
).
How much is y1(0.1) ?
Valaszok. A: 27.8292 , B: 31.2522 , C: 35.0963 , D: 39.4131 , E: 44.2609
Feladat. 7. Let
A =
(1 83 8
).
How much is det(e0.5A
)?
Valaszok. A: 90.0171 , B: 101.089 , C: 113.523 , D: 127.487 , E: 143.167
Feladat. 8. Let a = −1, b = 0, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00511835 , B: 0.00574791 , C: 0.0064549 , D: 0.00724885 , E: 0.00814046
Feladat. 9. Let a = 25, t0 = 3. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: 0.103128 , B: 0.115813 , C: 0.130058 , D: 0.146055 , E: 0.164019
Feladat. 10. Let a = 3, b = 2, t0 = 1.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.199613 , B: 0.224165 , C: 0.251737 , D: 0.282701 , E: 0.317473
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 7, b = 4, c = 3, d = 5 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 6944 , B: 71
44 , C: 7344 , D: 75
44 , E: 7944
Feladat. 12. Let a = 7, b = 2, c = 7, d = 7, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 2.26416 , B: 2.54266 , C: 2.8554 , D: 3.20662 , E: 3.60103
14
14.
Feladat. 1. Let y = (ay − b)(c − y), where a = 4, b = 2, c = 5 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 15 , B: 16 , C: 17 , D: 18 , E: 19
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 3, c = 2 . How much is u(2, 3)?
Valaszok. A: 0.33554 , B: 0.376812 , C: 0.42316 , D: 0.475208 , E: 0.533659
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.05, b = 2, c = 7 . How much is x13 ?
Valaszok. A: 34.334 , B: 38.5571 , C: 43.2997 , D: 48.6255 , E: 54.6064
Feladat. 4. Let y = tayb, y(2) = c, where a = 2, b = 1, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.85398 , B: 2.08202 , C: 2.33811 , D: 2.62569 , E: 2.94865
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 2, b = 3, c = 5 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 10 , B: 11 , C: 12 , D: 13 , E: 14
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(3 50 6
), ~y0 =
(126
).
How much is y1(0.1) ?
Valaszok. A: 20.9209 , B: 23.4942 , C: 26.384 , D: 29.6292 , E: 33.2736
Feladat. 7. Let
A =
(7 41 2
).
How much is det(e0.5A
)?
Valaszok. A: 71.3782 , B: 80.1577 , C: 90.0171 , D: 101.089 , E: 113.523
Feladat. 8. Let a = −3, b = 1, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00143374 , B: 0.00161009 , C: 0.00180813 , D: 0.00203053 , E: 0.00228028
Feladat. 9. Let a = 9, t0 = 6. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.198496 , B: −0.222911 , C: −0.250329 , D: −0.28112 , E: −0.315697
Feladat. 10. Let a = 3, b = 6, t0 = 3.75. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.132084 , B: 0.14833 , C: 0.166574 , D: 0.187063 , E: 0.210072
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 9, b = 6, c = 3, d = 6 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 8556 , B: 87
56 , C: 8956 , D: 93
56 , E: 9556
Feladat. 12. Let a = 2, b = 4, c = 3, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −2.32561 , B: −2.61166 , C: −2.9329 , D: −3.29364 , E: −3.69876
15
15.
Feladat. 1. Let y = (ay − b)(c − y), where a = 5, b = 3, c = 4 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 15 , B: 16 , C: 17 , D: 18 , E: 19
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 2, b = 5, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.0737175 , B: 0.0827848 , C: 0.0929673 , D: 0.104402 , E: 0.117244
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.08, b = 7, c = 4 . How much is x13 ?
Valaszok. A: 161.346 , B: 181.191 , C: 203.478 , D: 228.505 , E: 256.611
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 3, c = 1 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.09115 , B: 1.22536 , C: 1.37608 , D: 1.54534 , E: 1.73541
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 5, c = 9 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 35 , B: 36 , C: 37 , D: 38 , E: 39
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 80 12
), ~y0 =
(43
).
How much is y1(0.1) ?
Valaszok. A: 10.1978 , B: 11.4522 , C: 12.8608 , D: 14.4427 , E: 16.2191
Feladat. 7. Let
A =
(4 72 9
).
How much is det(e0.5A
)?
Valaszok. A: 418.211 , B: 469.651 , C: 527.418 , D: 592.29 , E: 665.142
Feladat. 8. Let a = −3, b = −1, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00552456 , B: 0.00620408 , C: 0.00696718 , D: 0.00782414 , E: 0.00878651
Feladat. 9. Let a = 36, t0 = 8. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0904096 , B: −0.10153 , C: −0.114018 , D: −0.128042 , E: −0.143792
Feladat. 10. Let a = 3, b = 4, t0 = −0.75. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: 0. , B: 0.0425 , C: 0.085 , D: 0.1275 , E: 0.17
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 4, b = 7, c = 3, d = 4 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 5138 , B: 53
38 , C: 5538 , D: 59
38 , E: 6138
Feladat. 12. Let a = 4, b = 7, c = 2, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 1.62339 , B: 1.82307 , C: 2.04731 , D: 2.29913 , E: 2.58192
16
16.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 3, c = 4 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 8 , B: 9 , C: 10 , D: 11 , E: 12
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 4, b = 3, c = 2 . How much is u(2, 3)?
Valaszok. A: 0.275751 , B: 0.309668 , C: 0.347757 , D: 0.390531 , E: 0.438567
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.06, b = 4, c = 1 . How much is x13 ?
Valaszok. A: 77.6615 , B: 87.2138 , C: 97.9411 , D: 109.988 , E: 123.516
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 2, c = 1 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 0.968263 , B: 1.08736 , C: 1.2211 , D: 1.3713 , E: 1.53997
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 2, b = 4, c = 6 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 14 , B: 15 , C: 16 , D: 17 , E: 18
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(3 70 10
), ~y0 =
(43
).
How much is y1(0.1) ?
Valaszok. A: 8.46367 , B: 9.5047 , C: 10.6738 , D: 11.9867 , E: 13.461
Feladat. 7. Let
A =
(1 51 9
).
How much is det(e0.5A
)?
Valaszok. A: 93.3154 , B: 104.793 , C: 117.683 , D: 132.158 , E: 148.413
Feladat. 8. Let a = −2, b = −1, c = 3, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0112103 , B: 0.0125891 , C: 0.0141376 , D: 0.0158765 , E: 0.0178293
Feladat. 9. Let a = 36, t0 = 2. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0894288 , B: −0.100429 , C: −0.112781 , D: −0.126653 , E: −0.142232
Feladat. 10. Let a = 4, b = 4, t0 = 4.75. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.155829 , B: 0.174995 , C: 0.19652 , D: 0.220692 , E: 0.247837
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 6, b = 4, c = 3, d = 7 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 6140 , B: 63
40 , C: 6740 , D: 69
40 , E: 7140
Feladat. 12. Let a = 5, b = 3, c = 5, d = 6, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 1.06481 , B: 1.19578 , C: 1.34286 , D: 1.50804 , E: 1.69352
17
17.
Feladat. 1. Let y = (ay − b)(c − y), where a = 4, b = 2, c = 4 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 10 , B: 11 , C: 12 , D: 13 , E: 14
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 2, b = 2, c = 4 . How much is u(2, 3)?
Valaszok. A: 0.163195 , B: 0.183268 , C: 0.20581 , D: 0.231125 , E: 0.259553
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.02, b = 6, c = 4 . How much is x13 ?
Valaszok. A: 83.0422 , B: 93.2564 , C: 104.727 , D: 117.608 , E: 132.074
Feladat. 4. Let y = tayb, y(2) = c, where a = 2, b = 1, c = 3 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.47637 , B: 2.78097 , C: 3.12303 , D: 3.50716 , E: 3.93854
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 2, b = 6, c = 7 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 24 , B: 25 , C: 26 , D: 27 , E: 28
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(3 50 7
), ~y0 =
(84
).
How much is y1(0.1) ?
Valaszok. A: 9.96883 , B: 11.195 , C: 12.572 , D: 14.1183 , E: 15.8549
Feladat. 7. Let
A =
(1 22 9
).
How much is det(e0.5A
)?
Valaszok. A: 117.683 , B: 132.158 , C: 148.413 , D: 166.668 , E: 187.168
Feladat. 8. Let a = −1, b = 2, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0340768 , B: 0.0382683 , C: 0.0429753 , D: 0.0482613 , E: 0.0541974
Feladat. 9. Let a = 4, t0 = 9. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.265133 , B: −0.297744 , C: −0.334367 , D: −0.375494 , E: −0.42168
Feladat. 10. Let a = 6, b = 3, t0 = 4.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.232746 , B: 0.261373 , C: 0.293522 , D: 0.329626 , E: 0.37017
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 5, b = 4, c = 3, d = 2 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 4936 , B: 53
36 , C: 5536 , D: 59
36 , E: 6136
Feladat. 12. Let a = 7, b = 7, c = 6, d = 5, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 4.64168 , B: 5.21261 , C: 5.85376 , D: 6.57377 , E: 7.38234
18
18.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 4, c = 5 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 10 , B: 11 , C: 12 , D: 13 , E: 14
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 5, b = 3, c = 4 . How much is u(2, 3)?
Valaszok. A: 0.229368 , B: 0.25758 , C: 0.289263 , D: 0.324842 , E: 0.364798
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.02, b = 8, c = 1 . How much is x13 ?
Valaszok. A: 74.6559 , B: 83.8386 , C: 94.1508 , D: 105.731 , E: 118.736
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 3, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.54018 , B: 1.72963 , C: 1.94237 , D: 2.18128 , E: 2.44958
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 7, c = 8 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 26 , B: 27 , C: 28 , D: 29 , E: 30
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 40 8
), ~y0 =
(33
).
How much is y1(0.1) ?
Valaszok. A: 4.49794 , B: 5.05119 , C: 5.67248 , D: 6.3702 , E: 7.15373
Feladat. 7. Let
A =
(6 73 8
).
How much is det(e0.5A
)?
Valaszok. A: 1096.63 , B: 1231.52 , C: 1383. , D: 1553.1 , E: 1744.14
Feladat. 8. Let a = 0, b = 2, c = 3, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00178104 , B: 0.0020001 , C: 0.00224612 , D: 0.00252239 , E: 0.00283264
Feladat. 9. Let a = 4, t0 = 3. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0986464 , B: −0.11078 , C: −0.124406 , D: −0.139708 , E: −0.156892
Feladat. 10. Let a = 6, b = 3, t0 = −0.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: −0.085 , B: −0.0425 , C: 0. , D: 0.0425 , E: 0.085
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 9, b = 5, c = 5, d = 3 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 3118 , B: 35
18 , C: 3718 , D: 41
18 , E: 4318
Feladat. 12. Let a = 4, b = 4, c = 3, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −2.86964 , B: −3.22261 , C: −3.61899 , D: −4.06412 , E: −4.56401
19
19.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 2, c = 2 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 4 , B: 5 , C: 6 , D: 7 , E: 8
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 3, c = 4 . How much is u(2, 3)?
Valaszok. A: 0.1788 , B: 0.200792 , C: 0.22549 , D: 0.253225 , E: 0.284372
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.05, b = 9, c = 8 . How much is x13 ?
Valaszok. A: 109.719 , B: 123.214 , C: 138.37 , D: 155.389 , E: 174.502
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 3, c = 1 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 0.686064 , B: 0.77045 , C: 0.865215 , D: 0.971637 , E: 1.09115
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 4, c = 5 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 1 , B: 2 , C: 3 , D: 4 , E: 5
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 60 8
), ~y0 =
(52
).
How much is y1(0.1) ?
Valaszok. A: 7.66002 , B: 8.6022 , C: 9.66027 , D: 10.8485 , E: 12.1828
Feladat. 7. Let
A =
(2 85 1
).
How much is det(e0.5A
)?
Valaszok. A: 2.81788 , B: 3.16448 , C: 3.55371 , D: 3.99082 , E: 4.48169
Feladat. 8. Let a = 2, b = 3, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00724885 , B: 0.00814046 , C: 0.00914174 , D: 0.0102662 , E: 0.0115289
Feladat. 9. Let a = 36, t0 = 1. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0328821 , B: −0.0369266 , C: −0.0414686 , D: −0.0465693 , E: −0.0522973
Feladat. 10. Let a = 3, b = 6, t0 = −2.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: −0.085 , B: −0.0425 , C: 0. , D: 0.0425 , E: 0.085
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 3, b = 5, c = 7, d = 4 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 136 , B: 17
6 , C: 196 , D: 23
6 , E: 256
Feladat. 12. Let a = 3, b = 2, c = 4, d = 4, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 0.123282 , B: 0.138446 , C: 0.155475 , D: 0.174598 , E: 0.196074
20
20.
Feladat. 1. Let y = (ay − b)(c − y), where a = 4, b = 4, c = 2 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 2 , B: 3 , C: 4 , D: 5 , E: 6
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 4, b = 4, c = 3 . How much is u(2, 3)?
Valaszok. A: 0.157294 , B: 0.176641 , C: 0.198368 , D: 0.222767 , E: 0.250168
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.05, b = 4, c = 3 . How much is x13 ?
Valaszok. A: 60.667 , B: 68.129 , C: 76.5089 , D: 85.9195 , E: 96.4876
Feladat. 4. Let y = tayb, y(2) = c, where a = 1, b = 3, c = 1 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 0.908874 , B: 1.02067 , C: 1.14621 , D: 1.28719 , E: 1.44551
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 4, b = 6, c = 9 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 26 , B: 27 , C: 28 , D: 29 , E: 30
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 80 11
), ~y0 =
(2521
).
How much is y1(0.1) ?
Valaszok. A: 58.3539 , B: 65.5314 , C: 73.5918 , D: 82.6436 , E: 92.8088
Feladat. 7. Let
A =
(3 32 9
).
How much is det(e0.5A
)?
Valaszok. A: 284.857 , B: 319.895 , C: 359.242 , D: 403.429 , E: 453.051
Feladat. 8. Let a = −2, b = 0, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00552456 , B: 0.00620408 , C: 0.00696718 , D: 0.00782414 , E: 0.00878651
Feladat. 9. Let a = 16, t0 = 9. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.247945 , B: −0.278442 , C: −0.31269 , D: −0.351151 , E: −0.394343
Feladat. 10. Let a = 2, b = 6, t0 = 5.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.104792 , B: 0.117682 , C: 0.132157 , D: 0.148412 , E: 0.166667
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 4, b = 9, c = 6, d = 3 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 7942 , B: 83
42 , C: 8542 , D: 89
42 , E: 9542
Feladat. 12. Let a = 3, b = 3, c = 2, d = 7, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 0.717251 , B: 0.805473 , C: 0.904546 , D: 1.01581 , E: 1.14075
21
21.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 4, c = 3 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 2 , B: 3 , C: 4 , D: 5 , E: 6
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 4, b = 5, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.120895 , B: 0.135765 , C: 0.152464 , D: 0.171217 , E: 0.192276
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.07, b = 3, c = 6 . How much is x13 ?
Valaszok. A: 74.881 , B: 84.0914 , C: 94.4346 , D: 106.05 , E: 119.094
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 3, c = 3 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 9.65837 , B: 10.8463 , C: 12.1804 , D: 13.6786 , E: 15.3611
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 4, b = 5, c = 6 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 5 , B: 6 , C: 7 , D: 8 , E: 9
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 40 6
), ~y0 =
(63
).
How much is y1(0.1) ?
Valaszok. A: 6.44701 , B: 7.23999 , C: 8.13051 , D: 9.13056 , E: 10.2536
Feladat. 7. Let
A =
(3 15 8
).
How much is det(e0.5A
)?
Valaszok. A: 172.775 , B: 194.026 , C: 217.891 , D: 244.692 , E: 274.789
Feladat. 8. Let a = 0, b = 1, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0309642 , B: 0.0347728 , C: 0.0390499 , D: 0.043853 , E: 0.0492469
Feladat. 9. Let a = 9, t0 = 1. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: 0.0372999 , B: 0.0418878 , C: 0.04704 , D: 0.0528259 , E: 0.0593235
Feladat. 10. Let a = 4, b = 6, t0 = 5.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.117637 , B: 0.132106 , C: 0.148356 , D: 0.166603 , E: 0.187095
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 6, b = 7, c = 3, d = 2 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 5146 , B: 53
46 , C: 5546 , D: 57
46 , E: 5946
Feladat. 12. Let a = 7, b = 7, c = 3, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 2.96241 , B: 3.32679 , C: 3.73598 , D: 4.19551 , E: 4.71156
22
22.
Feladat. 1. Let y = (ay − b)(c − y), where a = 2, b = 4, c = 5 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 3 , B: 4 , C: 5 , D: 6 , E: 7
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 5, c = 3 . How much is u(2, 3)?
Valaszok. A: 0.138158 , B: 0.155152 , C: 0.174236 , D: 0.195667 , E: 0.219734
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.03, b = 7, c = 8 . How much is x13 ?
Valaszok. A: 96.0035 , B: 107.812 , C: 121.073 , D: 135.965 , E: 152.688
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 3, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.73115 , B: 3.06708 , C: 3.44434 , D: 3.86799 , E: 4.34375
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 2, b = 5, c = 6 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 13 , B: 14 , C: 15 , D: 16 , E: 17
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(2 40 8
), ~y0 =
(99
).
How much is y1(0.1) ?
Valaszok. A: 13.4938 , B: 15.1536 , C: 17.0175 , D: 19.1106 , E: 21.4612
Feladat. 7. Let
A =
(7 12 7
).
How much is det(e0.5A
)?
Valaszok. A: 689.513 , B: 774.323 , C: 869.565 , D: 976.521 , E: 1096.63
Feladat. 8. Let a = −2, b = 0, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00102553 , B: 0.00115167 , C: 0.00129332 , D: 0.0014524 , E: 0.00163105
Feladat. 9. Let a = 36, t0 = 4. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.10657 , B: −0.119678 , C: −0.134399 , D: −0.15093 , E: −0.169494
Feladat. 10. Let a = 6, b = 3, t0 = −2.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! Howmuch is U(t0) ?
Valaszok. A: −0.085 , B: −0.0425 , C: 0. , D: 0.0425 , E: 0.085
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 6, b = 6, c = 6, d = 6 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 11544 , B: 117
44 , C: 11944 , D: 123
44 , E: 12544
Feladat. 12. Let a = 4, b = 3, c = 3, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 0.0742391 , B: 0.0833706 , C: 0.0936251 , D: 0.105141 , E: 0.118073
23
23.
Feladat. 1. Let y = (ay − b)(c − y), where a = 3, b = 2, c = 4 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 10 , B: 11 , C: 12 , D: 13 , E: 14
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 5, b = 5, c = 5 . How much is u(2, 3)?
Valaszok. A: 0.117689 , B: 0.132165 , C: 0.148421 , D: 0.166677 , E: 0.187178
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.04, b = 8, c = 8 . How much is x13 ?
Valaszok. A: 130.307 , B: 146.335 , C: 164.335 , D: 184.548 , E: 207.247
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 2, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 1.88604 , B: 2.11802 , C: 2.37854 , D: 2.6711 , E: 2.99965
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 3, b = 5, c = 6 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 6 , B: 7 , C: 8 , D: 9 , E: 10
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(3 60 7
), ~y0 =
(42
).
How much is y1(0.1) ?
Valaszok. A: 4.6472 , B: 5.2188 , C: 5.86071 , D: 6.58158 , E: 7.39112
Feladat. 7. Let
A =
(8 55 5
).
How much is det(e0.5A
)?
Valaszok. A: 418.211 , B: 469.651 , C: 527.418 , D: 592.29 , E: 665.142
Feladat. 8. Let a = 2, b = 3, c = 3, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0125891 , B: 0.0141376 , C: 0.0158765 , D: 0.0178293 , E: 0.0200223
Feladat. 9. Let a = 16, t0 = 9. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.155896 , B: −0.175072 , C: −0.196605 , D: −0.220788 , E: −0.247945
Feladat. 10. Let a = 4, b = 6, t0 = 4.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.104614 , B: 0.117481 , C: 0.131932 , D: 0.148159 , E: 0.166383
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 6, b = 2, c = 6, d = 5 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 10336 , B: 107
36 , C: 10936 , D: 113
36 , E: 11536
Feladat. 12. Let a = 2, b = 4, c = 7, d = 2, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −2.6139 , B: −2.93541 , C: −3.29647 , D: −3.70193 , E: −4.15727
24
24.
Feladat. 1. Let y = (ay − b)(c − y), where a = 4, b = 4, c = 2 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 3 , B: 4 , C: 5 , D: 6 , E: 7
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 3, b = 3, c = 2 . How much is u(2, 3)?
Valaszok. A: 0.236922 , B: 0.266063 , C: 0.298789 , D: 0.33554 , E: 0.376812
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.04, b = 5, c = 5 . How much is x13 ?
Valaszok. A: 81.4422 , B: 91.4596 , C: 102.709 , D: 115.342 , E: 129.529
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 3, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 3.06708 , B: 3.44434 , C: 3.86799 , D: 4.34375 , E: 4.87803
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 2, b = 3, c = 5 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 6 , B: 7 , C: 8 , D: 9 , E: 10
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 60 7
), ~y0 =
(52
).
How much is y1(0.1) ?
Valaszok. A: 9.54684 , B: 10.7211 , C: 12.0398 , D: 13.5207 , E: 15.1837
Feladat. 7. Let
A =
(8 78 5
).
How much is det(e0.5A
)?
Valaszok. A: 665.142 , B: 746.954 , C: 838.829 , D: 942.005 , E: 1057.87
Feladat. 8. Let a = −1, b = 0, c = 2, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.0390499 , B: 0.043853 , C: 0.0492469 , D: 0.0553043 , E: 0.0621067
Feladat. 9. Let a = 36, t0 = 1. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: −0.0369266 , B: −0.0414686 , C: −0.0465693 , D: −0.0522973 , E: −0.0587298
Feladat. 10. Let a = 2, b = 4, t0 = 3.25. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.249624 , B: 0.280328 , C: 0.314808 , D: 0.35353 , E: 0.397014
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 8, b = 2, c = 8, d = 9 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 18544 , B: 189
44 , C: 19144 , D: 193
44 , E: 19544
Feladat. 12. Let a = 4, b = 4, c = 5, d = 4, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: −3.52269 , B: −3.95598 , C: −4.44256 , D: −4.989 , E: −5.60264
25
25.
Feladat. 1. Let y = (ay − b)(c − y), where a = 2, b = 5, c = 3 . Find the smallest fixed point of the DE, thenwrite down the linearized ∆y = j∆y DE around that fixed point. The answer should be the number j.
Valaszok. A: 1 , B: 2 , C: 3 , D: 4 , E: 5
Feladat. 2. Let ∂tu(t, y) + (ay + b)∂yu(t, y) = 0, u(0, y) = 1c+y2
, where a = 5, b = 5, c = 2 . How much is u(2, 3)?
Valaszok. A: 0.264345 , B: 0.29686 , C: 0.333374 , D: 0.374379 , E: 0.420427
Feladat. 3. Let xn+1 = axn + b, x0 = c, where a = 1.04, b = 7, c = 9 . How much is x13 ?
Valaszok. A: 131.374 , B: 147.532 , C: 165.679 , D: 186.057 , E: 208.943
Feladat. 4. Let y = tayb, y(2) = c, where a = 3, b = 3, c = 2 . What is the prediction of Heun’s method fory(2.01)-re ?
Valaszok. A: 2.73115 , B: 3.06708 , C: 3.44434 , D: 3.86799 , E: 4.34375
Feladat. 5. Letd
dt~y =
((y1 − a)(y2 − b)
y1 − c
),
where a = 1, b = 3, c = 4 . Find the fixed point of the DE, the write down the linearized ddt~∆y = J ~∆y differential
equation around the fized point. As the answer, compute the sum of the squares of the entries of J .
Valaszok. A: 8 , B: 9 , C: 10 , D: 11 , E: 12
Feladat. 6. Let ddt~y = A~y, ~y(0) = ~y0, where
A =
(4 80 11
), ~y0 =
(2621
).
How much is y1(0.1) ?
Valaszok. A: 75.0836 , B: 84.3189 , C: 94.6901 , D: 106.337 , E: 119.416
Feladat. 7. Let
A =
(5 43 9
).
How much is det(e0.5A
)?
Valaszok. A: 774.323 , B: 869.565 , C: 976.521 , D: 1096.63 , E: 1231.52
Feladat. 8. Let a = −3, b = 3, c = 4, and let ∂tφ = c∂xxφ, φ(t, x) = φ(t, x+ 2π), φ(0, x) = χ[a,b](x) ifx ∈ (−π, π]. If
φ(t, x) =∑n∈Z
cn(t) · einx
√2π,
then how much is |c3(0.1)| ?
Valaszok. A: 0.00188305 , B: 0.00211466 , C: 0.00237477 , D: 0.00266686 , E: 0.00299489
Feladat. 9. Let a = 25, t0 = 8. Find the retarded solution of the G′′(t) + aG(t) = δ(t) DE! How much is G(t0) ?
Valaszok. A: 0.149023 , B: 0.167353 , C: 0.187937 , D: 0.211053 , E: 0.237013
Feladat. 10. Let a = 5, b = 5, t0 = 4.75. Find the retarded solution of the aU ′(t) + bU(t) = θ(t) DE! How muchis U(t0) ?
Valaszok. A: 0.157216 , B: 0.176553 , C: 0.19827 , D: 0.222657 , E: 0.250044
Feladat. 11. Legyen y′′(t) + ay′(t) + by(t) = et + 5, y(0) = c, y′(0) = d, ahol a = 8, b = 5, c = 9, d = 8 .Compute the Y (s) Laplace transform of y(t). How much is Y (2) ?
Valaszok. A: 20150 , B: 203
50 , C: 20750 , D: 209
50 , E: 21150
Feladat. 12. Let a = 2, b = 7, c = 4, d = 3, and let ∂ttφ = ∂xxφ, φ(t, 0) = φ(t, π) = 0, φ(0, x) =a sin(bx), φ(0, x) = c sin(dx). How much is φ(1, 2) ?
Valaszok. A: 1.28323 , B: 1.44107 , C: 1.61832 , D: 1.81737 , E: 2.04091
26
1 B E A E E E A B B B C B2 B A A E A B C A C C E D3 D E B D D B A E A E C E4 B E A D C B C C B E A A5 A C D C A B B B D B B D6 D E C E A C B C B D B A7 D B C C D E D D C E A A8 C C C D A D B C B B D C9 C C D D C E B E C E E D10 B C C E B B D B E D E A11 E D D D B D D A E E A B12 D A A A E E C A A D E C13 E B E B C A A D C D B C14 D A D B A A C D C C A A15 C C A A C B E B D A A D16 B D A B D B E E A E D D17 E C B C C D C C D C B C18 B A E D A C A D D C D C19 A B E E E C E A D C B E20 C E C B A C D B A E C D
27
21 D C A A A D D D C D E D22 D C C B E C E A D C A A23 A D B C E E E D E E D D24 B D B A E A A B C A A A25 A C A B C A D E A C B B
28