1M Questions on OrdinaryDifferential Equations of Higher-Order

1. The particular integral (P.I) of \((D^3 + 4D)y = \sin 2x + 1\) is:

1. \(\frac{x}{4} \sin 2x + \frac{x}{4}\)
2. \(\frac{-x}{8} \sin 2x + \frac{x}{4}\)
3. \(\frac{x}{8} \sin 2x + \frac{x}{8}\)
4. \(\frac{-x}{4} \sin 2x + \frac{x}{8}\)

Answer: b) \(\frac{-x}{8} \sin 2x + \frac{x}{4}\)

2. The particular integral (P.I) of \((D^3 - 1)y = \cos^2 \frac{x}{2}\) is:

1. \(\frac{-1}{2} - \frac{1}{4} ( \sin x + \cos x) \)
2. \(\frac{1}{2} + \frac{1}{4} ( \sin x + \cos x) \)
3. \(\frac{-1}{4} + \frac{1}{4} ( \cos x + \sin x)\)
4. \(\frac{1}{4} + \frac{1}{4} ( \sin x + \cos x)\)

Answer: a) \(\frac{-1}{2} - \frac{1}{4} ( \sin x + \cos x) \)

3. \(\frac{1}{(D+1)^2}x =\)

1. \(x - 2\)
2. \(x + 2\)
3. \(x - 1\)
4. \(x + 1\)

Answer: a) \(x - 2\)

4. The complementary function (C.F) of \((D^4 + 4)y = e^x \tan x\) is:

1. \(e^x[C_1\cos x+C_2\sin x]+e^{-x}[C_3\cos x+C_4\sin x]\)
2. \(e^x[C_1\cos x+C_2\sin x]+e^{-x}[C_3\cos 2x+C_4\sin 2x]\)
3. \(e^x[C_1\cos x-C_2\sin x]+e^{-x}[C_3\cos x+C_4\sin x]\)
4. \(e^x[C_1\cos x+C_2\sin x]+e^{-x}[C_3\cos 2x-C_4\sin 2x]\)

Answer: a) \(e^x[C_1\cos x+C_2\sin x]+e^{-x}[C_3\cos x+C_4\sin x]\)

5. The complete solution of \((D^2 - 2D + 5)^2 y = 0\) is:

1. \(e^x (C_1 \cos 2x + C_2 \sin 2x) + e^x (C_3 \cos 2x + C_4 \sin 2x)\)
2. \(e^x (C_1 \cos 2x + C_2 \sin 2x) + x e^x (C_3 \cos 2x + C_4 \sin 2x)\)
3. \(e^x (C_1 \cos 2x + C_2 \sin 2x) + e^{-x} (C_3 \cos 2x + C_4 \sin 2x)\)
4. \(e^x (C_1 \cos 2x + C_2 \sin 2x) + x e^{-x} (C_3 \cos 2x + C_4 \sin 2x)\)

Answer: b) \(e^x (C_1 \cos 2x + C_2 \sin 2x) + x e^x (C_3 \cos 2x + C_4 \sin 2x)\)

6. \(\frac{1}{(D-2)(D-3)} e^x =\)

1. \(\frac{e^x}{2}\)
2. \(\frac{e^x}{4}\)
3. \(\frac{2e^x}{2}\)
4. \(\frac{2e^x}{4}\)

Answer: a) \( \frac{e^x}{2}\)

7. The general solution of \((D^2 + 4)y = \sin 3x\) is:

1. \(C_1 \cos 2x + C_2 \sin 2x - \frac{1}{5} \sin 3x\)
2. \(C_1 \cos 2x + C_2 \sin 2x + \frac{1}{7} \sin 3x\)
3. \(C_1 \cos 2x + C_2 \sin 2x + \frac{1}{5} \sin 3x\)
4. \(C_1 \cos 2x + C_2 \sin 2x + \frac{1}{11} \sin 3x\)

Answer: a) \(C_1 \cos 2x + C_2 \sin 2x - \frac{1}{5} \sin 3x\)

8. The particular integral (P.I) of \((D^3 - 1)y = x^3\) is:

1. \(-x^3+6\)
2. \(-x^3-6\)
3. \(-x^3+4\)
4. \(-x^3-4\)

Answer: b) \(-x^3-6\)

9. The particular integral (P.I) of \((D+1)^2 y = e^{-x}\) is:

1. \(\frac{x^2}{2} e^{-x}\)
2. \(\frac{x}{2} e^{-x}\)
3. \(\frac{x^2}{2} e^{x}\)
4. \(\frac{x}{2} e^{x}\)

Answer: a) \(\frac{x^2}{2} e^{-x}\)

10. The particular integral (P.I) of \((D^2 + 4)y = \cos 2x\) is:

1. \(\frac{x}{4} \sin 2x\)
2. \(\frac{x}{4} \cos 2x\)
3. \(\frac{x}{2} \sin 2x\)
4. \(\frac{x}{2} \cos 2x\)

Answer: a) \(\frac{x}{4} \sin 2x\)

11. The particular integral (P.I) of \((D^2 - 2D + 4)y = e^x \cos x\) is:

1. \(\frac{e^x \cos x}{2}\)
2. \(\frac{e^x \sin x}{2}\)
3. \(\frac{e^x (\cos x + \sin x)}{2}\)
4. \(\frac{e^x (\cos x - \sin x)}{2}\)

Answer: a) \(\frac{e^x \cos x}{2}\)

12. The Wronskian of \(e^x \cos x\), \(e^x \sin x\) is:

1. \(e^{2x}\)
2. \(e^x\)
3. \(e^{3x}\)
4. \(e^{4x}\)

Answer: a) \(e^{2x}\)

13. The particular integral (P.I) of \((D^3 + d)y = \cos x\) is:

1. \(\frac{\sin x}{2}\)
2. \(\frac{\cos x}{2}\)
3. \(\frac{\sin x}{4}\)
4. \(\frac{\cos x}{4}\)

Answer: d) \(\frac{\cos x}{4}\)

14. The particular integral (P.I) of \((D + 2)y = 2^x\) is:

1. \(\frac{-\cos x}{2}\)
2. \(\frac{\cos x}{2}\)
3. \(\frac{\cos 2x}{2}\)
4. \(\frac{-\cos 2x}{2}\)

Answer: a) \(\frac{-\cos x}{2}\)

15. \(y = e^{-x} \left( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x \right) + c_3 e^{2x}\) is the general solution of:

1. \( (D^3 - 8)y = 0 \)
2. \( (D^3 + 8)y = 0 \)
3. \( (D^3 - 4)y = 0 \)
4. \( (D^3 + 4)y = 0 \)

Answer: a) \( (D^2 - 8)y = 0 \)

16. The complementary function (C.F) of \( x^2 y'' - xy' + 2y = x \) is:

1. \( x[C_1 \sin(\log x)+ C_2 \sin(\log x)] \)
2. \( x[C_1 \cos(\log x)+ C_2 \cos(\log x)] \)
3. \( x[C_1 \cos(\log x)+ C_2 \sin(\log x)] \)
4. \( x[C_1 \sin(\log x)+ C_2 \cos(\log x)] \)

Answer: c) \( x[C_1 \cos(\log x)+ C_2 \sin(\log x)] \)

17. The general solution of \( \frac{d^2 y}{dx^2} - 4 \frac{dy}{dx} + 6y = 0 \) is:

1. \( y = e^{2x}[C_1 \cos \sqrt{2}x+C_2 \sin \sqrt{2}x] \)
2. \( y = e^{x}[C_1 \sin \sqrt{2}x+C_2 \sin \sqrt{2}x] \)
3. \( y = e^{2x}[C_1 \cos \sqrt{2}x+C_2 \cos \sqrt{2}x] \)
4. \( y = e^{x}[C_1 \sin \sqrt{2}x+C_2 \cos \sqrt{2}x] \)

Answer: a) \( y = e^{2x}[C_1 \cos \sqrt{2}x+C_2 \sin \sqrt{2}x] \)

18. \(\frac{1}{(D-1)^5} e^x =\)

1. \(\frac{x^5}{120} e^x\)
2. \(\frac{x^4}{24} e^x\)
3. \(\frac{x^5}{24} e^x\)
4. \(\frac{x^4}{120} e^x\)

Answer: a) \(\frac{x^5}{120} e^x\)

19. If \( e^{(1+i)x} \) is a solution of the differential equation \( y'' + \alpha y' + \beta y = 0 \), where \( \alpha \) and \( \beta \) are real, then:

1. \( \alpha = 2 \), \( \beta = 2 \)
2. \( \alpha = -2 \), \( \beta = 2 \)
3. \( \alpha = 2 \), \( \beta = -2 \)
4. \( \alpha = -2 \), \( \beta = -2 \)

Answer: b) \( \alpha = -2 \), \( \beta = 2 \)

20. If the roots of an auxiliary equation of a 4th order differential equation are \( 1 \pm i, 1 \pm i \), then its complementary function (C.F) is:

1. \( e^x (C_1 \cos x + C_2 \sin x) + e^x (C_3 \cos x + C_4 \sin x) \)
2. \( e^x (C_1 \cos x + C_2 \sin x) + x e^x (C_3 \cos x + C_4 \sin x) \)
3. \( e^x (C_1 \cos x + C_2 \sin x) + e^{-x} (C_3 \cos x + C_4 \sin x) \)
4. \( e^x (C_1 \cos x + C_2 \sin x) + x e^{-x} (C_3 \cos x + C_4 \sin x) \)

Answer: b) \( e^x (C_1 \cos x + C_2 \sin x) + x e^x (C_3 \cos x + C_4 \sin x) \)

21. If \( \alpha \pm \sqrt{\beta} \) are the roots of the auxiliary equation of a second-order differential equation, then its complementary function (C.F) is:

1. \( e^{\alpha x} (C_1 \cos \sqrt{\beta} x + C_2 \sin \sqrt{\beta} x) \)
2. \( e^{\alpha x} (C_1 \cosh \sqrt{\beta} x + C_2 \sinh \sqrt{\beta} x) \)
3. \( e^{\alpha x} (C_1 \cos \beta x + C_2 \sin \beta x) \)
4. \( e^{\alpha x} (C_1 \cosh \beta x + C_2 \sinh \beta x) \)

Answer: a) \( e^{\alpha x} (C_1 \cos \sqrt{\beta} x + C_2 \sin \sqrt{\beta} x) \)

22. If \( f(D) = D^2 + 5 \), then \( \frac{1}{f(D)} \sin 2x = \):

1. \( \sin 2x \)
2. \( \sin x \)
3. \( \cos 2x \)
4. \( \cos x \)

Answer: a) \( \sin 2x \)

23. The particular integral (P.I) of \( (D^2 - 2D + 2)y = \log 4 \) is:

1. \( \frac{\log 4}{2} \)
2. \( \frac{\log 4}{4} \)
3. \( \frac{\log 4}{6} \)
4. \( \frac{\log 4}{8} \)

Answer: a) \( \frac{\log 4}{2} \)

24. The reduced form \( (x^2 D^2 + xD + 7)y = \frac{2}{x} \) converted to a linear differential equation with constant coefficients is:

1. \( \frac{d^2 y}{dt^2} + 7y = 2e^{-t} \)
2. \( \frac{d^2 y}{dt^2} + 7y = 2e^{t} \)
3. \( \frac{d^2 y}{dt^2} + 7y = 2e^{-2t} \)
4. \( \frac{d^2 y}{dt^2} + 7y = 2e^{2t} \)

Answer: a) \( \frac{d^2 y}{dt^2} + 7y = 2e^{-t} \)

25. The particular value of \( \frac{1}{(D-1)^4} e^x \sin x = \):

1. \( e^x \sin x \)
2. \( e^x \cos x \)
3. \( e^x \sin 2x \)
4. \( e^x \cos 2x \)

Answer: a) \( e^x \sin x \)

26. The differential equation governing Simple Harmonic Motion is:

1. \( \frac{d^2 x}{dt^2} + \omega^2 x = 0 \)
2. \( \frac{d^2 x}{dt^2} - \omega^2 x = 0 \)
3. \( \frac{d^2 x}{dt^2} + \omega x = 0 \)
4. \( \frac{d^2 x}{dt^2} - \omega x = 0 \)

Answer: a) \( \frac{d^2 x}{dt^2} + \omega^2 x = 0 \)

27. The substitution to reduce Legendre’s linear equation \( (ax + b)^3 y''' + (ax + b)^2 y'' + y = f(ax + b) \) into a linear equation with constant coefficients is:

1. \( t = \ln(ax + b) \)
2. \( t = ax + b \)
3. \( t = (ax + b)^2 \)
4. \( t = \frac{1}{ax + b} \)

Answer: a) \( t = \ln(ax + b) \)

28. The periodic time of the motion described by the differential equation \( (D^2 + \mu)x = 0 \) is:

1. \( \frac{2\pi}{\sqrt{\mu}} \)
2. \( \frac{\pi}{\sqrt{\mu}} \)
3. \( \frac{2\pi}{\mu} \)
4. \( \frac{\pi}{\mu} \)

Answer: a) \( \frac{2\pi}{\sqrt{\mu}} \)

29. The periodic time of the motion described by the differential equation \( \frac{d^2 x}{dt^2} + 4x = 0 \) is:

1. \( \pi \)
2. \( \frac{\pi}{2} \)
3. \( 2\pi \)
4. \( \frac{\pi}{4} \)

Answer: b) \( \frac{\pi}{2} \)

30. A condenser of capacity \( C \) discharged through an inductance \( L \) and resistance \( R \) in series and the differential equation governing this system is:

1. \( L \frac{d^2 q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0 \)
2. \( L \frac{d^2 q}{dt^2} + R \frac{dq}{dt} - \frac{q}{C} = 0 \)
3. \( L \frac{d^2 q}{dt^2} - R \frac{dq}{dt} + \frac{q}{C} = 0 \)
4. \( L \frac{d^2 q}{dt^2} - R \frac{dq}{dt} - \frac{q}{C} = 0 \)

Answer: a) \( L \frac{d^2 q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0 \)

31. The complete solution of \( (D-1)^2 y = 0 \) is:

1. \( y = C_1 e^x + C_2 x e^x \)
2. \( y = C_1 e^x + C_2 e^{-x} \)
3. \( y = C_1 e^{-x} + C_2 x e^x \)
4. \( y = C_1 e^x + C_2 x \)

Answer: a) \( y = (C_1 + C_2 x) e^x \)

32. The particular integral (P.I) of \( \frac{1}{D^2 - 1} e^x \) is:

1. \( \frac{x e^x}{2} \)
2. \( \frac{e^x}{2} \)
3. \( \frac{x e^x}{4} \)
4. \( \frac{e^x}{4} \)

Answer: a) \( \frac{x e^x}{2} \)

33. The particular integral (P.I) of \( (D^2 - 1) y = \sin x \cos x \) is:

1. \( \frac{-1}{10}\sin x \)
2. \( \frac{-1}{10}\sin 2x \)
3. \( \frac{-1}{10}\cos 2x \)
4. \( \frac{-1}{10}\cos x \)

Answer: b) \( \frac{-1}{10}\sin 2x \)

34. If the auxiliary equation of \( (x^2 D^2 - 3xD - 5) y = x \log x \) is \( m^2 + \alpha m + \beta = 0 \), then:

1. \( \alpha = -4 \), \( \beta = 5 \)
2. \( \alpha = 4 \), \( \beta = -5 \)
3. \( \alpha = -4\), \( \beta = -5 \)
4. \( \alpha = 4 \), \( \beta = 5 \)

Answer: c) \( \alpha = -4\), \( \beta = -5 \)

35. The complete solution of \( (D^3 - D^2) y = 0 \) is:

1. \( y = C_1 + C_2 x + C_3 e^x \)
2. \( y = C_1 + C_2 e^x + C_3 x e^x \)
3. \( y = C_1 + C_2 x + C_3 x^2 \)
4. \( y = C_1 + C_2 e^x + C_3 e^{-x} \)

Answer: a) \( y = C_1 + C_2 x + C_3 e^x \)

36. The particular integral (P.I) of \( (D^3 - D) y = 2e^x \) is:

1. \( xe^{-x} \)
2. \( x^2e^x \)
3. \( xe^{x} \)
4. \( x^2e^{-x} \)

Answer: c) \( xe^x \)

37. The complementary function (C.F) of \( (D^2 - 3D + 2) y = \cos 3x \) is:

1. \( C_1 e^x + C_2 e^{2x} \)
2. \( C_1 e^x + C_2 e^{-2x} \)
3. \( C_1 e^{-x} + C_2 e^{2x} \)
4. \( C_1 e^{-x} + C_2 e^{-2x} \)

Answer: a) \( C_1 e^x + C_2 e^{2x} \)

38. The particular integral (P.I) of \( (D + 2)(D - 1)^2 y = e^{-2x} \) is:

1. \( \frac{e^{-2x}}{18} \)
2. \( \frac{e^{-2x}}{9} \)
3. \( \frac{e^{-2x}}{6} \)
4. \( \frac{e^{-2x}}{3} \)

Answer: b) \( \frac{e^{-2x}}{9} \)

39. The particular integral (P.I) of \( (D^2 + D) y = x^2 \) is:

1. \( \frac{x^3}{3} - x^2 + 2x \)
2. \( \frac{x^3}{3} + x^2 - 2x \)
3. \( \frac{x^3}{3} - x^2 - 2x \)
4. \( \frac{x^3}{3} + x^2 + 2x \)

Answer: a) \( \frac{x^3}{3} - x^2 + 2x \)

40. The particular integral (P.I) of \( (D - 1)^2 y = x^2 e^x \) is:

1. \( \frac{x^4 e^x}{12} \)
2. \( \frac{x^4 e^x}{24} \)
3. \( \frac{x^4 e^x}{6} \)
4. \( \frac{x^4 e^x}{3} \)

Answer: a) \( \frac{x^4 e^x}{12} \)

41. Two linearly independent solutions of \((D^2 - 4)y = 0\) are:

1. \( e^{2x} \) and \( e^{-2x} \)
2. \( e^{2x} \) and \( x e^{2x} \)
3. \( e^{2x} \) and \( e^{x} \)
4. \( e^{2x} \) and \( x e^{x} \)

Answer: a) \( e^{2x} \) and \( e^{-2x} \)

42. The complementary function (C.F) of \(\left(D^3 - D\right)y = 2e^x\) is:

1. \( C_1 + C_2 e^x + C_3 e^{-x} \)
2. \( C_1 + C_2 x + C_3 e^x \)
3. \( C_1 + C_2 e^x + C_3 x e^x \)
4. \( C_1 + C_2 x + C_3 x^2 \)

Answer: a) \( C_1 + C_2 e^x + C_3 e^{-x} \)

43. The particular integral (P.I) of \((D^2 + 5)y = \cos x \sin x\) is:

1. \( \frac{\sin 2x}{10} \)
2. \( \frac{\sin 2x}{5} \)
3. \( \frac{\sin 2x}{2} \)
4. \( \frac{\sin 2x}{4} \)

Answer: c) \( \frac{\sin 2x}{2} \)

44. The complementary function (C.F) of \(\left(D^3 + 3D^2 + 3D + 1\right)y = \sin 2x\) is:

1. \( (C_1 + C_2 x + C_3 x^2) e^{-x} \)
2. \( C_1 e^{-x} + C_2 e^{x} + C_3 e^{2x} \)
3. \( C_1 + C_2 x + C_3 x^2 \)
4. \( C_1 e^{-x} + C_2 x e^{-x} + C_3 x^2 e^{-x} \)

Answer: a) \( (C_1 + C_2 x + C_3 x^2) e^{-x} \)

45. The value of \(\frac{1}{(xD + 1)}x^{-1}\) where \(D = \frac{d}{dx}\) is:

1. \( \frac{1}{x^2} \log x \)
2. \( \frac{1}{x} \log x \)
3. \( \frac{1}{x^3} \log x \)
4. \( \frac{1}{x^4} \log x \)

Answer: b) \( \frac{1}{x} \log x \)

46. The particular integral (P.I) of \((D - 2)^2y = 8\sin 2x\) is:

1. \( \cos 2x \)
2. \( \sin 2x \)
3. \( \cos x \)
4. \(\sin 2x \)

Answer: a) \( \cos 2x \)

47. Two linearly independent solutions of D.E \((D^2 + 2D + 1)y = 0\) are:

1. \( e^{-x} \) and \( x e^{-x} \)
2. \( e^{-x} \) and \( e^{x} \)
3. \( e^{-x} \) and \( e^{-2x} \)
4. \( e^{-x} \) and \( x e^{x} \)

Answer: a) \( e^{-x} \) and \( x e^{-x} \)

48. The particular integral (P.I) of \((D - 2)^2y = xe^{2x}\) is:

1. \( \frac{x^3 e^{2x}}{6} \)
2. \( \frac{x^2 e^{2x}}{2} \)
3. \( \frac{x e^{2x}}{2} \)
4. \( \frac{x^3 e^{2x}}{3} \)

Answer: a) \( \frac{x^3 e^{2x}}{6} \)

49. \(\frac{1}{D^3 + 4D}\sin 2x =\)

1. \( x\frac{-\sin 2x}{8} \)
2. \( x\frac{\sin 2x}{4} \)
3. \( x\frac{\sin 2x}{2} \)
4. \( x\frac{\sin 2x}{1} \)

Answer: a) \( x\frac{-\sin 2x}{8} \)

50. \(\frac{1}{D^2 - k^2}\left(\frac{e^{\int dx} + e^{-\int dx}}{2}\right)y =\)

1. \( \frac{sinhx}{1-k^2} \)
2. \( \frac{coshx}{1-k^2} \)
3. \( \frac{e^{-x}-e^x}{1-k^2} \)
4. \( \frac{e^{x}+e^{-x}}{1-k^2} \)

Answer: b) \( \frac{coshx}{1-k^2} \)

51. The complete solution of \( y'' + 2py' + (p^2 + q^2)y = 0 \) is:

1. \( y = e^{-px} (C_1 \cos qx + C_2 \sin qx) \)
2. \( y = e^{px} (C_1 \cos qx + C_2 \sin qx) \)
3. \( y = e^{-px} (C_1 \cosh qx + C_2 \sinh qx) \)
4. \( y = e^{px} (C_1 \cosh qx + C_2 \sinh qx) \)

Answer: a) \( y = e^{-px} (C_1 \cos qx + C_2 \sin qx) \)

52. The set of linearly independent solutions of the D.E \(\left( D^4 - D^2 \right)y = 0\) is:

1. \( 1, x, e^x, e^{-x} \)
2. \( 1, x, \cos x, \sin x \)
3. \( 1, x, e^x, x e^x \)
4. \( 1, x, \cosh x, \sinh x \)

Answer: a) \( 1, x, e^x, e^{-x} \)

53. \( \frac{1}{(D-1)(D-2)^2} e^{2x} = \)

1. \( \frac{x^2 e^{2x}}{2} \)
2. \( \frac{x e^{2x}}{2} \)
3. \( \frac{e^{2x}}{2} \)
4. \( \frac{x^2 e^{2x}}{4} \)

Answer: a) \( \frac{x^2 e^{2x}}{2} \)

54. The complete solution of \(\left( x^2 D^2 + 2xD - 2 \right)y = 0\) is:

1. \( y = C_1 x + C_2 x^{-2} \)
2. \( y = C_1 x^2 + C_2 x^{-1} \)
3. \( y = C_1 x + C_2 x^2 \)
4. \( y = C_1 x^2 + C_2 x^{-2} \)

Answer: a) \( y = C_1 x + C_2 x^{-2} \)

55. The reduced form \((1+x)^2 y'' + (x + 1)y + y = 2\sin[\log(1+x)]\) converted to a linear differential equation with constant coefficients by substituting \(1 + x = e^z\) is:

1. \( \frac{d^2 y}{dz^2} + y = 2\sin z \)
2. \( \frac{d^2 y}{dz^2} - y = 2\sin z \)
3. \( \frac{d^2 y}{dz^2} + y = 2\cos z \)
4. \( \frac{d^2 y}{dz^2} - y = 2\cos z \)

Answer: a) \( \frac{d^2 y}{dz^2} + y = 2\sin z \)

56. The complete solution of \(\left( x^2 D^2 - 5xD + 9 \right)y = 0\) is:

1. \( y = C_1 x^3 + C_2 x^{-3} \)
2. \( y = C_1 x^3 + C_2 x^3 \log x \)
3. \( y = C_1 x^3 + C_2 x^{-3} \log x \)
4. \( y = C_1 x^3 + C_2 x^3 \)

Answer: b) \( y = C_1 x^3 + C_2 x^3 \log x \)

57. The complete solution of \(\left( D^3 + 2D^2 + D \right)y = 0\) is:

1. \( y = C_1 + C_2 e^{-x} + C_3 x e^{-x} \)
2. \( y = C_1 + C_2 e^{x} + C_3 x e^{x} \)
3. \( y = C_1 + C_2 e^{-x} + C_3 e^{x} \)
4. \( y = C_1 + C_2 e^{x} + C_3 e^{-x} \)

Answer: a) \( y = C_1 + C_2 e^{-x} + C_3 x e^{-x} \)

58. The reduced form \( (x^2 D^2 - 3xD + 4)y = \sin(\log x) + 1 \) converted to a linear equation with constant coefficients by substituting \(x = e^z\) is:

1. \( \frac{d^2 y}{dz^2} - 4 \frac{dy}{dz} + 4y = \sin z + 1 \)
2. \( \frac{d^2 y}{dz^2} - 4 \frac{dy}{dz} + 4y = \sin z \)
3. \( \frac{d^2 y}{dz^2} - 4 \frac{dy}{dz} + 4y = 1 \)
4. \( \frac{d^2 y}{dz^2} - 4 \frac{dy}{dz} + 4y = \cos z \)

Answer: a) \( \frac{d^2 y}{dz^2} - 4 \frac{dy}{dz} + 4y = \sin z + 1 \)

59. The solution of \(\left( D^2 + 1 \right)y = 0\) satisfying the initial conditions \(y(0) = 1, y\left( \frac{\pi}{2} \right) = 2\) is:

1. \( y = \cos x + 2\sin x \)
2. \( y = \cos x + \sin x \)
3. \( y = 2\cos x + \sin x \)
4. \( y = 2\cos x + 2\sin x \)

Answer: a) \( y = \cos x + 2\sin x \)

60. The set of linearly independent solutions for the DE \(\left( x^2 D^2 - xD + 1 \right)y = 0\) is:

1. \( x, x \log x \)
2. \( x, x^2 \)
3. \( x, \log x \)
4. \( x, x^{-1} \)

Answer: a) \( x, x \log x \)