# Mathematics Questions and Answers – General and Particular Solutions of Differential Equation

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses 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=2ex
c) y=e-2x
d) y=e2x

Explanation: 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

Explanation: 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

Explanation: 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

Explanation: 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

Explanation: 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

Explanation: 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

Explanation: 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

Explanation: The number of arbitrary constants in a general solution of a nth 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

Explanation: The number of arbitrary constants for a particular solution of nth order differential equation is always zero.

10. The function y=3 cos⁡x is a solution of the function $$\frac{d^2 y}{dx^2}-3\frac{dy}{dx}$$=0.
a) True
b) False

Explanation: 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.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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