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
View Answer

Answer: c
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.
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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
View Answer

Answer: a
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
View Answer

Answer: d
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.
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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
View Answer

Answer: c
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
View Answer

Answer: a
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.
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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
View Answer

Answer: a
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
View Answer

Answer: b
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.
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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
View Answer

Answer: c
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
View Answer

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

Answer: b
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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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter