Linear differential equations with constant coefficients mcq

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Linear differential equations with constant coefficients mcq

Linear differential equations with constant coefficients mcq questions and answers for multiple choice questions test. Linear differential equations (LDE) with constant coefficients is a topic of Engineering Mathematics 3 (M3) for online exams of various universities. We are trying to provide you m3 linear differential equation mcq pdf for download the mcqs.

1.If the roots m₁, m₂, m₃, …, m_n of auxiliary equation φ(D) = 0 are real and distinct, then solution of φ(D) y = 0 is …………
(A) c₁em1x + c₂em2x + … + c_nemnx
(B) c₁ cos m₁x + c₂ cos m₂x + … + c_n cos m_nx
(C) m₁ec1x + m₂ec2x + … + mncnx
(D) c₁ sin m₁x + c₂ sin m₂x + … + c_n sin m_nx

c1em1x + c2em2x + … + cnemnx

2.The roots m₁, m₂, m₃ …, m_n of auxiliary equation φ(D) = 0 are real. If two of these roots are repeated say m₁ = m₂ and theremaining roots m₃, m₄, …, m_n are distinct then solution of φ(D) y = 0 is …………
(A) c₁em1x + c2m2x + … + cnmnx
(B)(c₁x + c₂) cos m₁x + c₃ cos m₃x + x … + c_n cos m_nx
(C)(c₁x + c₂) em1x + c₃em3x + … + c_nemnx
(D)(c₁x + c₂) sin m₁x + c₃ sin m₃x + … + c_n sin m_nx

(c1x + c2) em1x + c3em3x + … + cnemnx

3.The roots m₁, m₂, m₃ …, m_n of auxiliary equation φ(D) = 0 are real. If three of these roots are repeated, say, m₁ = m₂ = m₃ andthe remaining roots m₄, m₅, … m_n are distinct then solution of φ(D) y = 0 is …………
(A) c₁em1x + c2m2x + … + c_n emnx
(B)(c₁x² + c₂x + c₃) em1x + c₄em4x + … + c_nemnx
(C)(c₁x² + c₂x + c₃) cos m₁x + c₄ cos m₄x + … + c_n cos m_nx
(D)(c₁x² + c₂x + c₃) sin m₁x + c₄ sin m₄x + … + c_n sin m_nx

(c1x2 + c2x + c3) em1x + c4em4x + … + cnemnx

4.If m₁ = α + iβ and m₂ = α – iβ are two complex roots of auxiliary equation of second order DE φ(D) y = 0 then it’s solution is…………
(A) eβx [c₁ cos αx + c₂ sin αx]
(B) eαx [(c₁x + c₂) cos βx + (c₃x + c₄) sin βx]
(C) c₁eαx + c₂eβx
(D) eαx [c₁ cos βx + c₂ sin βx]

eαx [c1 cos βx + c2 sin βx]

5.If the complex roots m₁ = α + iβ and m₂ = α – iβ of auxiliary equation of fourth order DE φ(D) y = 0 are repeated twice thenit’s solution is …………
(A) eβx [c₁ cos αx + c₂ sin αx]
(B) eαx [(c₁x + c₂) cos βx + (c₃x + c₄) sin βx]
(C)(c₁x + c₂) eαx + (c₃x + c₄) eβx
(D) eαx [c₁ cos βx + c₂ sin βx]d²ydy

eαx [(c1x + c2) cos βx + (c3x + c4) sin βx]

6.The solution of differential equation d²y/dx² – 5 dy/dx + 6y = 0 is …………
(A) c₁e^2x + c₂e^{– 3x}
(B) c₁e^-2x + c₂e^3x
(C) c₁e^-2x + c₂e^-3x
(D) c₁e^2x + c₂e^3xd2ydy

c1e2x + c2e3xd2ydy

7.The solution of differential equation d²y/dx² – 5 dy/dx – 6y = 0 is …………
(A) c₁e^-x + c₂e^6x
(B) c₁e^-2x + c₂e^-3x
(C) c₁e^3x + c₂e^2x
(D) c₁e^-3x + c₂e^-2xd²y dy

c1e-x + c2e6x

8.The solution of differential equation 2 d²y/dx² – dy/dx – 10y = 0 is …………55x
(A) c₁e^2x + c₂e2 x
(B) c₁e^-2x + c₂e- 2532 ^x
(C) c₁e^-2x + c₂e2 x
(D) c₁e^-2x + c₂ed²y

c1e-2x + c2e2 x

9.The solution of differential equation d²y/dx² – 4y = 0 is …………
(A)(c₁x + c₂) e^2x
(B) c₁e^4x + c₂e^-4x
(C) c₁ cos 2x + c₂ sin 2x
(D) c₁e^2x + c₂e^-2xd²y dy

c1e2x + c2e-2xd2y dy

10. The solution of differential equation d²y/dx² – dy/dx – 2y = 0 is …………
(A) c₁e^2x + c₂e^x
(B) c₁e^2x + c₂e^-x
(C) c₁e^-2x + c₂e^x
(D) c₁e^-2x + c₂e^-x(A.1)

c1e2x + c2e-x

Engineering Mathematics 3 mcq

11. The solution of differential equation 2 d²y/dx² – 5 dy/dx + 3y = 0 is …………33×32 ^x2 ^x
(A) c₁e^x + c₂e2 x
(B) c₁e^2x + c₂e^-3x
(C) c₁e^-x + c₂e
(D)c₁e2 + c2ed²ydy

c1ex + c2e2 x

12. The solution of differential equation d²y/dx² + 2 dy/dx + y = 0 is …………
(A) c₁e^2x + c₂e^x
(B) c₁e^x + c₂e^-x
(C)(c₁x + c₂) e^-x
(D)(c₁x + c₂) e^xd²ydy

(c1x + c2) e-x

13. The solution of differential equation 4 d²y/dx² – 4 dy/dx + y = 0 is …………xx- 2
(A) c₁e² + c₂e
(B)(c₁ + c₂x) e^-2xx
(C) c₁ cos 2x + c₂ sin 2x
(D)(c₁ + c₂x) e²d²ydy

(c1 + c2x) e2d2ydy

14. The solution of differential equation d²y/dx² – 4 dy/dx + 4y = 0 is …………
(A)(c₁x + c₂) e^2x
(B)(c₁x + c₂) e^-2x
(C) c₁e^4x + c₂e^-4x
(D)c₁e^2x + c₂e^-2xd²ydy

(c1x + c2) e2x

15. The solution of differential equation d²y/dx² + 6 dy/dx + 9y = 0 is …………
(A) c₁e^-6x + c₂e^-9x
(B)(c₁x + c₂) e^-3x
(C)(c₁x + c₂) e^3x
(D)c₁e^3x + c₂e^2xd²y

(c1x + c2) e-3x

16. The solution of differential equation d²y/dx² + y = 0 is …………
(A) c₁e^x + c₂e^-x
(B)(c₁x + c₂) e^-x
(C) c₁ cos x + c₂ sin x
(D)ex (c₁ cos x + c₂ sin x)d²y

c1 cos x + c2 sin x

17. The solution of differential equation d²y/dx² + 9y = 0 is …………
(A) c₁ cos 2x + c₂ sin 2x
(B)(c₁x + c₂) e^-3x
(C) c₁e^3x + c₂e^-3x
(D)c₁ cos 3x + c₂ sin 3xd²ydy

c1 cos 3x + c2 sin 3xd2ydy

18. The solution of differential equation d²y/dx² + 6 dy/dx + 10y = 0 is …………
(A) e^-3x (c₁ cos x + c₂ sin x)
(B) e^x (c₁ cos 3x + c₂ sin 3x)
(C) c₁e^5x + c₂e^2x
(D) e^x (c₁ cos x + c₂ sin x)d²y dy

e-3x (c1 cos x + c2 sin x)

19. The solution of differential equation d²y/dx² + dy/dx + y = 0 is …………33
(A) e^x (c₁ cos x + c₂ sin x)
(B) e^x/2c₁ cosx + c₂ sinx221–2 ^x33
(C) ec₁ cosx + c₂ sinx
(D) c₁e^x + c₂e^-x22d²ydy

ec1 cosx + c2 sinx

20. The solution of differential equation 4 d²y/dx² + 4 dy/dx + 5y = 0 is …………
(A) e^-x (c₁ cos 2x + c₂ sin 2x)
(B) e^-x/2 [c₁ cos x + c₂ sin x]
(C) e^-2x (c₁ cos x + c₂ sin x)
(D) c₁e^-4x+ c₂e^-5xd³yd²ydy

e-x/2 [c1 cos x + c2 sin x]

22. The solution of differential equation d³y/dx³ – 7 dy/dx – 6y = 0 is …………
(A) c₁e^x + c₂e^2x + c₃e^3x
(B) c₁e^-x + c₂e^-2x + c₃3^6x
(C) c₁e^-x + c₂e^2x + c₃e^x
(D) c₁e^-x + c₂e^-2x + c₃e^3xd³yd²y dy

c1e-x + c2e-2x + c3e3xd3yd2y dy

23. The solution of differential equation d³y/dx³ + 2 d²y/dx² + dy/dx = 0 is …………
(A) c₁ + e^x (c₂x + c₃)
(B) c₁ + e^-x (c₂x + c₃)
(C) e^-x (c₂x + c₃)
(D) c₁ + c₂e^x + c₃e^-xd³yd²ydy

c1 + e-x (c2x + c3)

24. The solution of differential equation………… d³y/dx³ – 5 d²y/dx² + 8 dy/dx – 4y = 0 is
(A) c₁e^x + (c₂x + c₃) e^2x
(B) c₁e^x + c₂e^2x+c₃e^3x
(C)(c₂x + c₃) e^2x
(D) c₁e^-x + (c₂x + c₃) e^-2xd³ydy

c1ex + (c2x + c3) e2x

25. The solution of differential equation d³y/dx³ – 4 dy/dx = 0 is …………
(A) c₁e^2x + c₂e^-2x
(B) c₁ + c₂ cos 2x + c₃sin 2x
(C) c₁e^x + c₂e^-2x + c₃e^-3x
(D) c₁ + c₂e^2x + c₃e^-2xd³y

c1 + c2e2x + c3e-2xd3y

26. The solution of differential equation d³y/dx³ + y = 0 is …………13311
(A) c₁e^x + e^xc₂ cos
(B) c₁e^-x + e2 xc₂ cos2 x + c3 sin2 x2 x + c3 sin 2 x133
(C) c₁e^-x + e2 xc₂ cos
(D)(c₁ + c₂x + c₃x²) e^-x2 x + c3 sin2 xd³ydy

c1e-x + e2 xc2 cos

27. The solution of differential equation d³y/dx³ + 3 dy/dx = 0 is …………
(A) c₁ + c₂ cos x + c₃ sin x
(B) c₁ + c₂ cos3x + c₃ sin3x
(C) c₁ + c₂e^3x + c₃e^–3x
(D) c₁ cos x + c₂ sin xd³y d²ydy

c1 + c2 cos3x + c3 sin3x

28. The solution of differential equation………… d³y/dx³ + d²y/dx² – 2 dy/dx + 12y = 0 is
(A) c₁e^-3x + e^x (c2 cos3x + c₃ sin3x)
(B) c₁e^-3x + (c₂ cos 3x + c₃ sin 3x)
(C) c₁e^3x + e^-x (c2 cos3x + c₃ sin3x)
(D) c₁e^-x + c₂e^-3x + c₃e3xd

c1e-3x + ex (c2 cos3x + c3 sin3x)

29. The solution of differential equation (D³ – D² + 3D + 5) y = 0 where D =dx is …………
(A) c₁e^-x + e^x (c₂ cos 2x + c₃ sin 2x)
(B) c₁e^-x + (c₂ cos 3x + c₃ sin 3x)
(C) c₁e^x + e^-x (c₂ cos 2x + c₃ sin 2x)
(D) c₁e^-x + c₂e^-2x + c₃e^-3xd³y d²ydy

c1e-x + ex (c2 cos 2x + c3 sin 2x)

30. The solution of differential equation d³y/dx³ – d²y/dx² + 4 dy/dx – 4y = 0 is …………
(A)(c₁ + c₂x) e^-2x + c₃e^-x
(B) c₁e^x + c₂ cos 4x + c₃ sin 4x
(C) c₁e^x + c₂ cos 2x + c₃ sin 2x
(D) c₁e^x + c₂e^2x + c₃e^-2xd⁴y

c1ex + c2 cos 2x + c3 sin 2x

Mathematics 3 mcq Computer Engineering

31. The solution of differential equation d⁴y/dx⁴ – y = 0 is …………
(A)(c₁x + c₂) e^-x + c₃ cos x + c₄ sin x
(B)(c₁x + c₂) cos x + (c₃x + c₄) sin x
(C)(c₁ + c₂x + c₃x² + c₄x³) e^x
(D) c₁e^x + c₂e^-x + c₃ cos x + c₄ sin xd

c1ex + c2e-x + c3 cos x + c4 sin xd

32. The solution of differential equation (D⁴ + 2D² + 1) y = 0 where D = dx is …………
(A)(c₁x + c₂) e^x + (c₃x + c₄) e^-x
(B)(c₁x + c₂) cos x + (c₃x + c₄) sin x
(C) c₁e^x + c₂e^-x + c₃ cos x + c₄ sin x
(D)(c₁x + c₂) cos 2x + (c₃x + c₄) sin 2x

(c1x + c2) cos x + (c3x + c4) sin x

33. The solution of differential equation (D² + 9)² y = 0, where D =(2)dx is …………
(A)(c₁x + c₂) e^3x + (c₃x + c₄) e^-3x
(B)(c₁x + c₂) cos 3x + (c₃x + c₄) sin 3x
(C)(c₁x + c₂) cos 9x + (c₃x + c₄) sin 9x
(D)(c₁x + c₂) cos x + (c₃x + c₄) sin xd⁴yd²y

(c1x + c2) cos 3x + (c3x + c4) sin 3x

34. The solution of differential equation d⁴y/dx⁴ + 8 d²y/dx² + 16y = 0 is …………
(A) c₁e^2x + c₂e^-x + c₃e^x + c₄e^-2x
(B)(c₁x + c₂) e^2x + (c₃x + c₄) e^-2x
(C)(c₁x + c₂) cos 4x + (c₃x + c₄) sin 4x
(D)(c₁x + c₂) cos 2x + (c₃x + c₄) sin 2xd⁶yd⁴yd²y

(c1x + c2) cos 2x + (c3x + c4) sin 2xd6yd4yd2y

35. The solution of differential equation d⁶y/dx⁶ + 6 d⁴y/dx⁴ + 9 d²y/dx² = 0 is …………
(A) c₁x + c₂ + (c₃x + c₄) cos3x + (c₃x + c₆) sin3x
(B) c₁x + c₂ + (c₃x + c₄) cos 3x + (c₅x + c₆) sin 3x
(C)(c₁x + c₂) cos3x + (c₃x + c₄) sin3x
(D) c₁x + c₂ + (c₃x + c₄) e^3x.

c1x + c2 + (c3x + c4) cos3x + (c3x + c6) sin3x

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