250+ TOP MCQs on General and Particular Solutions of Differential Equation | Class 12 Maths

Mathematics Multiple Choice Questions on “General and Particular Solutions of Differential Equation”.

1. Which of the following functions is the solution of the differential equation (frac{dy}{dx})+2y=0?
a) y=-2e^-x
b) y=2e^x
c) y=e^-2x
d) y=e^2x
Answer: c
Clarification: Consider the function y=e^-2x
Differentiating both sides w.r.t x, we get
(frac{dy}{dx}=-2e^{-2x})
(frac{dy}{dx})=-2y
⇒(frac{dy}{dx})+2y=0.

2. The function y=8 sin⁡2x is a solution of the differential equation (frac{d^2 y}{dx^2})+4y=0.
a) True
b) False
Answer: a
Clarification: The given statement is true.
Consider the function y=8 sin⁡2x
Differentiating w.r.t x, we get
(frac{dy}{dx})=16 cos⁡2x –(1)
Differentiating (1) w.r.t x, we get
(frac{d^2 y}{dx^2})=-32 sin⁡2x
(frac{d^2 y}{dx^2})=-4(8 sin⁡2x )=-4y
⇒(frac{d^2 y}{dx^2})+4y=0.

3. Which of the following functions is a solution for the differential equation xy’-y=0?
a) y=4x
b) y=x²
c) y=-4x
d) y=2x
Answer: d
Clarification: Consider the function y=2x
Differentiating w.r.t x, we get
y’=(frac{dy}{dx})=2
Substituting in the equation xy’-y, we get
xy’-y=x(2)-2x=2x-2x=0
Therefore, the function y=2x is a solution for the differential equation xy’-y=0.

4. Which of the following differential equations has the solution y=3x²?
a) (frac{d^2 y}{dx^2})-6x=0
b) (frac{dy}{dx})-3x=0
c) x (frac{d^2 y}{dx^2})–(frac{dy}{dx})=0
d) (frac{d^2 y}{dx^2}-frac{3dy}{dx})=0
Answer: c
Clarification: Consider the function y=3x²
Differentiating w.r.t x, we get
(frac{dy}{dx})=6x –(1)
Differentiating (1) w.r.t x, we get
(frac{d^2 y}{dx^2})=6
∴(frac{xd^2 y}{dx^2}-frac{6dy}{dx})=6x-6x=0
Hence, the function y=3x² is a solution for the differential equation x (frac{d^2 y}{dx^2})-6 (frac{dy}{dx})=0.

5. Which of the following functions is a solution for the differential equation y”+6y=0?
a) y=5 cos⁡3x
b) y=5 tan⁡3x
c) y=cos⁡3x
d) y=6 cos⁡3x
Answer: a
Clarification: Consider the function y=5 cos⁡3x
Differentiating w.r.t x, we get
y’=(frac{dy}{dx})=-15 sin⁡3x
Differentiating again w.r.t x, we get
y”=(frac{d^2 y}{dx^2})=-30 cos⁡3x
⇒y”+6y=0.
Hence, the function y=5 cos⁡3x is a solution for the differential equation y”+6y=0.

6. Which of the following functions is a solution for the differential equation (frac{dy}{dx})-14x=0?
a) y=7x²
b) y=7x³
c) y=x⁷
d) y=14x
Answer: a
Clarification: Consider the function y=7x²
Differentiating w.r.t x, we get
(frac{dy}{dx})=14x
∴(frac{dy}{dx})-14x=0
Hence, the function y=7x² is a solution for the differential equation (frac{dy}{dx})-14x=0

7. Which of the following differential equations given below has the solution y=log⁡x?
a) (frac{d^2 y}{dx^2})-x=0
b) (frac{d^2 y}{dx^2}+(frac{dy}{dx})^2)=0
c) (frac{d^2 y}{dx^2})–(frac{dy}{dx})=0
d) x (frac{d^2 y}{dx^2})-log⁡x=0
Answer: b
Clarification: Consider the function y=log⁡x
Differentiating w.r.t x, we get
(frac{dy}{dx}=frac{1}{x} )–(1)
Differentiating (1) w.r.t x, we get
(frac{d^2 y}{dx^2}=-frac{1}{x^2} )
∴(frac{d^2 y}{dx^2}+(frac{dy}{dx})^2=-frac{1}{x^2}+(frac{1}{x})^2)
=-(frac{1}{x^2}+frac{1}{x^2})=0.

8. How many arbitrary constants will be there in the general solution of a second order differential equation?
a) 3
b) 4
c) 2
d) 1
Answer: c
Clarification: The number of arbitrary constants in a general solution of a n^th order differential equation is n.
Therefore, the number of arbitrary constants in the general solution of a second order D.E is 2.

9. The number of arbitrary constants in a particular solution of a fourth order differential equation is __________________
a) 1
b) 0
c) 4
d) 3
Answer: b
Clarification: The number of arbitrary constants for a particular solution of n^th order differential equation is always zero.

10. The function y=3 cos⁡x is a solution of the function (frac{d^2 y}{dx^2}-3frac{dy}{dx})=0.
a) True
b) False
Answer: b
Clarification: The given statement is false.
Given differential equation: (frac{d^2 y}{dx^2})-3 (frac{dy}{dx})=0 –(1)
Consider the function y=3 cos⁡x
Differentiating w.r.t x, we get
(frac{dy}{dx})=-3 sin⁡x
Differentiating again w.r.t x, we get
(frac{d^2 y}{dx^2})=-3 cos⁡x
Substituting the values of (frac{dy}{dx}) and (frac{d^2 y}{dx^2}) in equation (1), we get
(frac{d^2 y}{dx^2})-3 (frac{dy}{dx})=-3 cos⁡x-3(-3 sin⁡x)
=9 sin⁡x-3 cos⁡x≠0.
Hence, y=3 cos⁡x, is not a solution of the equation (frac{d^2 y}{dx^2})-3 (frac{dy}{dx})=0.