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Differential Equations
One Mark Questions
PREPARED BY:R.RAJENDRAN. M.A., M. Sc., M. Ed.,K.C.SANKARALINGA NADAR HR. SEC. SCHOOL,CHENNAI-21
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Choose the Correct Answer 1. The integrating factor of is
(a) log x (b) x2
(c) ex (d) x
2. If cos x is an integrating factor of the differential
equation then P is
(a) – cot x (b) cot x
(c) tan x (d) – tan x
xex
y
dx
dy 42
QPydx
dy
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Choose the Correct Answer
3. The integrating factor of dx + x dy = e– y sec2y dy is
(a) ex (b) e–x
(c) ey (d) e–y
4. The integrating factor of is
(a) ex (b) log x
(c) 1/x (d) e–x
2
2
log
1
xyxxdx
dy
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Choose the Correct Answer 5. The solution of where m < 0 is
(a) x = cemy (b) x = ce–m y
(c) x = my + c (d) x = c
6. y = cx – c2 is the general solution of the differential equation
(a) (y’)2 – xy’ + y = 0 (b) y’’ = 0
(c) y’ = c (d) (y’)2 + xy’ + y = 0
0mxdx
dy
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Choose the Correct Answer 7. The differential equation of is
(a) of order 2 and degree 1 (b) of order 1 and degree 2
(c) of order 1 and degree 6 (d) of order 1 and degree 3
8. The differential equation of all non-vertical lines in a plane is
(a) (b)
(c) (d)
xydx
dy
3/1
2
5
0dx
dy
mdx
dy
02
2
dx
yd
mdx
yd
2
2
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Choose the Correct Answer 9. The differential equation of all circles with center at the
origin is
(a) xdy + ydx = 0 (b) xdy – ydx = 0
(c) xdx + ydy = 0 (d) xdx – ydy = 0
10. The integrating factor of the differential equation
is
(a) P dx (b) Q dx
(c) e Q dx (d) e P dx
QPydx
dy
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Choose the Correct Answer 11. The complementary function of (D2 + 1)y = e 2x is is
(a) (Ax + B)ex (b) A cosx + B sinx
(c) (Ax + B)e2x (d) (Ax + B)e– x
12. The particular integral of (D2 – 4D + 4) y = e2x is
(a)(x2/2)e2x (b) xe2x
(c) xe– 2x (d) (x/2)e– 2x
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Choose the Correct Answer 13. The differential equation of the family of lines y = mx is
(a) (b) ydx – xdy = 0
(c) (d) ydx + xdy = 0
14. The degree of the differential equation
(a) 1 (b) 2
(c) 3 (d) 6
mdx
dy
02
2
dx
yd
2
23/1
1dx
yd
dx
dy
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Choose the Correct Answer 15. The degree of the differential
equation where c is a constant
(a) 1 (b) 3
(c) – 2 (d) 2
16. The differential equation satisfied by all the straight lines in xy plane is (a) = a constant (b)
(c) (d)
3
3
3/23
1
dxyd
dxdy
c
0dx
dyy 0
2
2
ydx
yddx
dy0
2
2
dx
yd
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Choose the Correct Answer 17. The amount present in a radio active element
disintegrates at a rate proportional to its amount. The differential equation corresponding to the above statement is (k is negative)
(a) (b) (c) (d)
18. If then
(a) 2xy + y2 – x2 = c (b) x2 + y2 – x + y = c
(c) x2 + y2 – 2xy = c (d) x2 – y2 – 2xy = c
ktdt
dp
p
k
dt
dp kp
dt
dpkt
dt
dp
yx
yx
dx
dy
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Choose the Correct Answer 19. If y = kex then its differential equation is
(a) (b)
(c) (d)
20. The differential equation formed by eliminating a and b from y = ae3x + be–3x is
(a) (b)
(c) (d)
kydx
dyy
dx
dy
xedx
dy 0 kydx
dy
02
2
aydx
yd
092
2
dx
dy
dx
yd
092
2
ydx
yd
092
2
ydx
yd
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Choose the Correct Answer 21. If f’(x) = x and f(1) = 2 then f(x) is
(a) (b)
(c) (d)
22. The differential equation formed by eliminating A and B from y = ex (A cosx + B sinx) is
(a) y2 + y1 = 0 (b) y2 – y1 = 0
(c) y2 – 2y1 + 2y = 0 (d) y2 – 2y1 – 2y = 0
)2(3
2 xx
)2(3
2xx
)2(2
3xx
)2(2
3 xx
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Choose the Correct Answer 23. On putting y = vx, the homogeneous differential
equation x2dy + y(x + y)dx = 0 becomes (a) x dv + (2v + v2)dx = 0 (b) v dx + (2x + x2)dv = 0(c) v2dx – (x + x2)dv = 0 (d) v dx + (2x + x2)dx = 0
24. The integrating factor of the differential equation
(a) secx (b) cosx
(c) etanx (d) cotx
xxydx
dycostan
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Choose the Correct Answer 25. The PI of (3D2 + D – 14)y = 13 e2x is
(a) 26xe2x (b) 13xe2x
(c) xe2x (d) (x2/2)e2x
26. The PI of the differential equation f(D)y = eax where f(D) = (D – a) g(D), g(a) 0 is
(a) meax (b)
(c) g(a)eax (d)
)(ag
eax
)(ag
xeax
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Choose the Correct Answer 27. The order and degree of the differential equation
is
(a) 1, 8 (b) 3, 1 (c) 3, 8 (d) 3, 2
28. The solution of is
(a) y = c ecosx (b) y = c e– sinx
(c) y = c e– cosx (d) y = c esinx
3
38
1dx
yd
dx
dy
0sin xydx
dy
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Choose the Correct Answer 29. The PI of (D2 + 1)y = 5 is
(a) 5 (b) 5/y
(c) x/5 (d) – 5
30. The integrating factor of the differential equation
(a) log(logy) (b)
(c) logy (d) 1/y
)log(log
1
y
is 0loglog yxdy
dxyy
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Choose the Correct Answer 31. The particular integral of (D2 – 3D + 2)y = ex is
(a) xex (b) – xex
(c) ex (d) – ex
32. The solution of the differential equation xdy + ydx is
(a) x2 y2 = c2 (b) y = cx2
(c) y = x2 + c (d) xy = c
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Choose the Correct Answer 33. The integrating factor of
(a) – log sinx (b) – sinx
(c) log sinx (d) cosecx
34. The order and degree of
(a) 1, 2 (b) 2, 1
(c) 2, 2 (d) 1, 1
is sincot xxydx
dy
are 62
2
2
ydx
dy
dx
yd
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Choose the Correct Answer 35. The integrating factor of
(a) sin2x (b) 1/log x
(c) 1/x (d) log x
36. The order and degree of
(a) 1, 4 (b) 2, 3
(c) 2, 4 (d) 1, 4
is 2sinlog xx
y
dx
dyx
are 03
2
22
4
ydx
ydx
dx
dyy
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Choose the Correct Answer
37. The order and degree of the differential equation
y’’ = (y – y’3) 2/3 are
(a) 2, 3 (b) 3, 3
(c) 3, 2 (d) 2, 2
38. The order and degree of the differential equation
y’ + (y’’)2 = (x + y’’) 2 are
(a) 1, 1 (b) 1, 2
(c) 2, 1 (d) 2, 2
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Choose the Correct Answer
39. The order and degree of the differential equation
y’ + (y’’)2 = x(x + y’’) 2 are
(a) 2, 2 (b) 2, 1
(c) 1, 2 (d) 1, 1
40. The order and degree of the differential equation
sin x (dx + dy) = cos x (dx - dy) are
(a) 1, 1 (b) 0, 0
(c) 1, 2 (d) 2, 1
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Choose the Correct Answer41. The differential equation corresponding to
y = ax2 + bx + c where a, b, c are arbitrary constants, is
(a) y’’’ + y’’ = 0 (b) y’’ = 2a
(c) y’’’ = 0 (d) y’’’ – y’’ = 0
42. In finding the differential equation corresponding to
y = emx where m is the arbitrary constant, then m is
(a) y / y’ (b) y’ / y
(c) y’ (d) y
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Choose the Correct Answer43. The order and degree of the differential equation
y’ + y2 = x are
(a) 2, 1 (b) 1, 1
(c) 1, 0 (d) 0, 1
44. The order and degree of the differential equation
y’’ + 3y’2 + y3 = 0 are
(a) 2, 2 (b) 2, 1
(c) 1, 2 (d) 3, 1
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Choose the Correct Answer45. The solution of a linear differential equation
where P and Q are functions of y is
(a) y(I.F) = (b) x(I.F) =
(c) y(I.F) = (d) x(I.F) =
QPxdy
dx
cdx (I.F)Q cdy (I.F)Q
cdy (I.F)Q cdx (I.F)Q
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Choose the Correct Answer46. The solution of a linear differential equation
where P and Q are functions of x is
(a) y(I.F) = (b) x(I.F) =
(c) y(I.F) = (d) x(I.F) =
QPydx
dy
cdx (I.F)Q cdy (I.F)Q
cdy (I.F)Q cdx (I.F)Q
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Choose the Correct Answer47. Identify the incorrect statement
(a) The order of a differential equation is the order of the
highest order derivative occurring in it
(b) The degree of the differential equation is the degree of the
highest order derivative which occurs in it (the derivatives are
free from radicals and fractions)
(c) is the first order first degree homogeneous
differential equation
(d) is a linear differential equation in x
),(
),(
2
1
yxf
yxf
dx
dy
xexydx
dy
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