Differential Equations

Question 1

1. A differential equation is an equation involving ___.
   A. Only constants B. Derivatives C. Polynomials D. Logarithms

Answer 1

✅ Answer: B – Derivatives
By definition, it contains one or more derivatives of an unknown function.

Question 2

2. Order of the DE (y’’ + 3y’ + 2y = 0) is ___.
   A. 1 B. 2 C. 3 D. 0

Answer 2

✅ Answer: B – 2
Highest order of derivative is second.

Question 3

3. Degree of ( (y’’’)^2 + y = 0) is ___.
   A. 1 B. 2 C. 3 D. 6

Answer 3

✅ Answer: B – 2
Derivative (y’’’) raised to power 2 → degree 2.

Question 4

4. The general solution of (dy/dx = 0) is ___.
   A. y = x B. y = C C. x = C D. y = 0

Answer 4

✅ Answer: B – y = C
Integration of 0 gives a constant.

Question 5

5. Differential equation of the family of straight lines (y = mx + c) is ___.
   A. y′ = m B. y″ = 0 C. y′ – m = 0 D. y = c

Answer 5

✅ Answer: B – y″ = 0
All lines have zero second derivative.

Question 6

6. (dy/dx = 3x^2) → y =?
   A. x³ + C B. x³/3 + C C. x³ – C D. x² + C

Answer 6

✅ Answer: A – x³ + C
Integrate 3x² → x³ + C.

Question 7

7. Solve (dy/dx = 2x + 1).
   A. y = x² + x + C B. y = x² + x C. y = x² – x + C D. y = 2x + C

Answer 7

✅ Answer: A – y = x² + x + C
Integrate term by term.

Question 8

8. A first-order linear DE has the form ___.
   A. y″ + P y′ + Q y = 0 B. dy/dx + P y = Q C. y = Px + Q D. y′ y = Q

Answer 8

✅ Answer: B – dy/dx + P y = Q
Standard form for integrating-factor method.

Question 9

9. Integrating factor (IF) for (dy/dx + P(x)y = Q(x)) is ___.
   A. e^(–∫P dx) B. e^(∫P dx) C. 1/P D. Q/P

Answer 9

✅ Answer: B – e^(∫P dx)
Multiply by IF to make LHS an exact derivative.

Question 10

10. A solution containing arbitrary constants is called ___.
    A. Particular solution B. General solution C. Singular solution D. Trivial solution

Answer 10

✅ Answer: B – General solution
Still includes constant of integration.

Question 11

11. The solution obtained after substituting given initial values is ___.
    A. General B. Particular C. Implicit D. Singular

Answer 11

✅ Answer: B – Particular solution
Constants are determined numerically.

Question 12

12. (dy/dx = ky) represents which law?
    A. Hooke’s B. Exponential growth/decay C. Ohm’s D. Newton’s Cooling

Answer 12

✅ Answer: B – Exponential growth/decay
Solution form y = C e^{kx}.

Question 13

13. Solve (dy/dx = 3y).
    A. y = 3x + C B. y = Ce^{3x} C. y = 3e^{x} D. y = Cx³

Answer 13

✅ Answer: B – y = Ce^{3x}
Separate and integrate: dy/y = 3dx.

Question 14

14. The solution of (y′ = –ky) describes ___.
    A. Cooling process B. Heating C. Linear motion D. Elasticity

Answer 14

✅ Answer: A – Cooling process
Form of Newton’s Law of Cooling.

Question 15

15. A DE that can be written as Mdx + Ndy = 0 is called ___.
    A. Exact B. Linear C. Separable D. Homogeneous

Answer 15

✅ Answer: A – Exact
It is exact if ∂M/∂y = ∂N/∂x.

Question 16

16. Condition for exactness of Mdx + Ndy = 0:
    A. ∂M/∂x = ∂N/∂y B. ∂M/∂y = ∂N/∂x C. M = N D. dM = dN

Answer 16

✅ Answer: B – ∂M/∂y = ∂N/∂x
Equality of mixed partials → exact.

Question 17

17. If (dy/dx = f(x)g(y)), it is called ___.
    A. Linear B. Homogeneous C. Separable D. Exact

Answer 17

✅ Answer: C – Separable
Variables can be separated as dy/g(y)=f(x)dx.

Question 18

18. Solve (dy/dx = x y).
    A. y = C e^{x²/2} B. y = x e^{C} C. y = C e^{x} D. y = x² + C

Answer 18

✅ Answer: A – y = C e^{x²/2}
Separate: dy/y = x dx → ln y = x²/2 + C.

Question 19

19. The complementary function (CF) of (y’’ + y = 0) is ___.
    A. C₁ e^{x} + C₂ e^{–x} B. C₁ cos x + C₂ sin x C. C₁ x + C₂ D. C₁ e^{ix} + C₂ e^{–ix}

Answer 19

✅ Answer: B – C₁ cos x + C₂ sin x
Characteristic eqn r² + 1 = 0 → r = ±i.

Question 20

20. For (y’’ – 4y = 0), CF = ___.
    A. C₁ cos 2x + C₂ sin 2x B. C₁ e^{2x} + C₂ e^{–2x} C. C₁ e^{x} + C₂ e^{–x} D. C₁ cosh x + C₂ sinh x

Answer 20

✅ Answer: B – C₁ e^{2x} + C₂ e^{–2x}
r² – 4 = 0 → r = ±2.

Question 21

21. Particular integral (PI) of (y’’ + y = \sin x) is ___.
    A. ½ sin x B. ½ cos x C. –½ cos x D. x sin x

Answer 21

✅ Answer: C – –½ cos x
Plug sin x forcing term into operator (1 + D²) PI = sin x.

Question 22

22. General solution of (y’’ + y = \sin x):
    A. C₁ cos x + C₂ sin x – ½ cos x B. C₁ cos x + C₂ sin x + ½ cos x C. C₁ sin x + C₂ cos x D. C₁ e^{x} + C₂ e^{–x}

Answer 22

✅ Answer: A – C₁ cos x + C₂ sin x – ½ cos x
General = CF + PI.

Question 23

23. If (y’’ + 4y′ + 4y = 0), roots of auxiliary eqn are ___.
    A. Distinct real B. Equal real C. Complex D. Imaginary

Answer 23

✅ Answer: B – Equal real
r² + 4r + 4 = 0 → ( r + 2 )² = 0.

Question 24

24. For equal roots r₁ = r₂ = –2, CF = ___.
    A. (C₁ + C₂ x) e^{–2x} B. C₁ e^{–2x} + C₂ e^{2x} C. C₁ cos 2x + C₂ sin 2x D. C₁ e^{x} + C₂ x e^{x}

Answer 24

✅ Answer: A – (C₁ + C₂ x) e^{–2x}
Equal roots → multiply by x for second term.

Question 25

25. The DE ((d²y/dx²) + 9y = 0) represents ___. A. Simple harmonic motion B. Exponential decay C. Uniform motion D. Projectile path

Answer 25

✅ Answer: A – Simple harmonic motion Characteristic roots ±3i → sinusoidal solution.