CLASS XII DIFFERENTIAL EQUATION DPP

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DIFFERENTIAL EQUATION

1. The integrating factor of the differential equation (x log x) dy/dx + y = 2

log x, is given by

(a) log (log x) (b) ex (c) log x (d) x

2. The general solution of the differential equation dy/dx = y/x is

(a) log y = kx (b) y = kx (c) xy = k (d) y = k log x

3. Integrating factor of the differential equation cos x dy/dx + y sin x = 1, is

(a) sin x (b) sec x (c) tan x (d) cos x

4. The degree of the differential equation (d2y/dx2)2 - (dy/dx)3 = y3, is

(a) 1/2 (b) 2 (c) 3 (d) 4

5. The degree of the differential equation

{5 + (dy/dx)2}(5/3) = x5 (d2y/dx2), is

(a) 4 (b) 2 (c) 3 (d) 10

6. The general solution of the differential equation

dy/dx + y cot x = cosec x, is

(a) x + y sin x = C (b) x + y cos x = C

7. The differential equation obtained on eliminating A and B from

y = A cos ωt + B sin ωt is

(a) y'' + y' = 0 (b) y'' - ω^2 y = 0

8. The equation of the curve whose slope is given by dy/dx = 2y/x;

x > 0, y > 0 and which passes through the point (1, 1) is

(a) x2 = y (b) y2 = x (c) x2 = 2y (d) y2 = 2x

9. The order of the differential equation whose general solution is given by

y = c₁ cos (2x+c₂) - (c₃ + c₄) ax + c₅ + c₆ sin (x - c₇) is

(a) 3 (b) 4 (c) 5 (d) 2

10. The solution of the differential equation dy/dx = (ax + g)/(by+f)

represents a circle when

(a) a = b (b) a = -b (c) a = -2b (d) a = 2b

11. The solution of the differential equation dy/dx + 2y/x = 0 with

y(1) = 1 is given by

(a) y = 1/x2 (b) x = 1/y2 (c) x = 1/y (d) y = 1/x

12. The solution of the differential equation dy/dx = y(x+1)/x = 0 is given

(a) y=xex+C (b) x = yex (c) y = x + C (d) xy = ex + C

13. The order of the differential equation satisfying √(1-x4) + √(1-y4) =

a(x²-y²) is

(a) 1 (b) 2 (c) 3 (d) 4

14. The solution of the differential equation y₁y₃ = y₂² is

(a) x = C1eC2y +C3 (b) y = C₁eC²x + C₃

15. The general solution of the differential equation

dy/dx + yg'(x) = g(x) g'(x), where g(x) is a given function of x, is

(a) g(x) + log (1 + y + g(x)) = C

(b) g(x) + log (1 + y - g(x)) = C

(d) none of these

16. The solution of the differential equation

dy/dx = 1 + x + y² + xy², y(0) = 0 is

(a) y² = exp (x + x²/2) - 1

(b) y² = 1 + C exp (x + x²/2)

(d) y = tan (x + x²/2)

17. The differential equation of the ellipse x²/a² + y²/b² = C is

(a) y''/y + y'/y - 1/x = 0

(b) y''/y + y'/y + 1/x = 0

(d) none of these

18. Solution of the differential equation dy/dx + y/x = sin x is

(a) x(y + cos x) = sin x + C

(b) x(y - cos x) = sin x + C

(d) none of these

19. The equation of the curve satisfying the differential equation

y(x + y³) dx = x(y³ - x) dy and passing through the point (1, 1) is

(a) y³ - 2x + 3x²y = 0

(b) y³ + 2x + 3x²y = 0

(d) none of these

20. The solution of the differential equation 2x dy/dx - y = 3 represents

(a) circles (b) straight lines

21. The solution of the differential equation x dy/dx = y + x tan y/x, is

(a) sin y/x = x + C (b) sin y/x = Cx

22. The differential equation satisfied by ax² + by² = 1 is

(a) xyy₂ + y₁² + yy₁ = 0

(b) xyy₂ + xy₁² - yy₁ = 0

23. The differential equation which represents the family of curves y =

eCx is

(a) y₁ = C²y (b) xy₁ - ln y = 0

24. Which of the following transformations reduce the differential

equation

dz / dx + (z/x) log z = (z/x) (log z)² into the form

du/dx + P (x) u = Q (x)

(a) u = log x (b) u = e²

25. The solution of the differential equation dy/dx = y/x + φ(y/x) is

(a) φ(y/x) = kx (b) x φ(y/x) = k

26. If m and n are the order and degree of the differential equation

(y₂)5 + 4(y₂)3 / y₃ = x³, then

(a) m = 3, n = 3 (b) m = 3, n = 2

27. The solution of the differential equation dy/dx + 1 = e(x+y), is

(a) (x + y) e(x+y) = 0 (b) (x + C) e(x+y) = 0

28. The solution of x² + y² dy/dx = 4, is

(a) x² + y² = 12x+C (b) x² + y² = 3x+C

29. The family of curves in which the subtangent at any point of a curve is

double the abscissa is given by

(a) x = Cy² (b) y = Cx² (c) x² = Cy² (d) y = Cx

30. The solution of the differential equation x dx + y dy = x² y dy - y² x dx,

(a) x² - 1 = C (1 + y²) (b) x² + 1 = C (1 - y²)

31. The solution of the differential equation (x² + 1) dy/dx + (y² + 1) = 0,

(a) y = 2 + x² (b) y = (1+x) / (1-x)

32. The differential equation x dy/dx - y = x², has the general solution

(a) y - x^3 = 2cx (b) 2y - x^3 = cx

33. The solution of the differential equation dy / dx - ky = 0, y(0) = 1

approaches to zero when x -> ∞, if

(a) k = 0 (b) k > 0 (c) k < 0 (d) none of these

34. The solution of the differential equation (1 + x2) dy / dx + 1 + y2 =

0, is

(a) tan-1 x - tan-1 y = tan-1 C

(b) tan-1 y - tan-1 x = tan-1 C

(d) tan-1 y + tan-1 x = tan-1 C

35. The solution of the differential equation dy / dx = (x2 + xy + y2) /

x2, is

(a) tan-1 (x / y) = log y + C

(b) tan-1 (y / x) = log x + C

(d) tan-1 (y / x) = log y + C

36. The differential equation dy / dx + Py = Qyn, n > 2 can be reduced to

linear form by substituting

(a) z = y(n-1) (b) z = yn

37. If p and q are the order and degree of the differential equation

dy / dx + x^3 d^2 y / dx^2 + xy = cos x, then

(a) p < q (b) p = q (c) p > q (d) none of these

38, Which of the following is the integrating factor of (x log x)

dy / dx + y = 2 log x?

(a) x (b) ex (c) log x (d) log (log x)

39. What is integrating factor of dy / dx + y sec x = tan x?

(a) sec x + tan x (b) log (sec x + tan x)

40. Integrating factor of the differential equation

cos x dy / dx + y sin x = 1, is

(a) cos x (b) tan x (c) sec x (d) sin x

41. The degree of the differential equation

(d2 y / dx2)3 + (dy / dx)2 + sin (dy / dx) + 1 = 0, is

(a) 3 (b) 2 (c) 1 (d) not defined

42. The order of the differential equation

2x2 d2 y / dx2 - 3 dy / dx + y = 0, is

(a) 2 (b) 1 (c) 0 (d) not defined

43. The number of arbitrary constants in the general solution of

differential equation of fourth order is

(a) 0 (b) 2 (c) 3 (d) 4

44. The number of arbitrary constants in the particular solution of a

differential equation of second order is (are)

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