Differential Equations

One Mark Questions

PREPARED BY:R.RAJENDRAN. M.A., M. Sc., M. Ed.,K.C.SANKARALINGA NADAR HR. SEC. SCHOOL,CHENNAI-21

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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