Differential Equations MCQs Question and Answer in Engineering Maths

31

The general solution of $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{y}} = 0$$ is

A) y = P cos x + Q sin x B) y = P cos x C) y = P sin x D) y = P sin² x

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Answer & Solution

Answer: Option A

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32

The solution for the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {{\text{x}}^2}{\text{y}}$$ with the condition that y = 1 at x = 0 is

A) $${\text{y}} = {{\text{e}}^{\frac{1}{{2{\text{x}}}}}}$$ B) $$\ln \left( {\text{y}} \right) = \frac{{{{\text{x}}^3}}}{3} + 4$$ C) $$\ln \left( {\text{y}} \right) = \frac{{{{\text{x}}^2}}}{2}$$ D) $${\text{y}} = {{\text{e}}^{\frac{{{{\text{x}}^3}}}{3}}}$$

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Answer: Option D

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33

The solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\text{ky}},\,{\text{y}}\left( 0 \right) = {\text{c}}$$ is

A) x = ce^-ky B) x = ke^cy C) y = ce^kx D) y = ce^-kx

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Answer & Solution

Answer: Option C

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34

The solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + {{\text{y}}^2} = 0$$ is

A) $${\text{y}} = \frac{1}{{{\text{x}} + {\text{c}}}}$$ B) $${\text{y}} = \frac{{ - {{\text{x}}^3}}}{3} + {\text{c}}$$ C) ce^x D) unsolvable as equation is non-linear

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Answer & Solution

Answer: Option A

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35

If y = 2x³ - 3x² + 3x - 10, the value of ∆³y will be (where, ∆ is forward differences operator)

A) 10 B) 11 C) 12 D) 13

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Answer: Option C

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36

Match List-I with List-II and select the correct answer:

List-I	List-II
A. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{y}}}{{\text{x}}}$$	1. Circles
B. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{y}}}{{\text{x}}}$$	2. Straight lines
C. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{x}}}{{\text{y}}}$$	3. Hyperbolas
D. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{x}}}{{\text{y}}}$$

A) a-2, b-3, c-3, d-1 B) a-1, b-3, c-2, d-1 C) a-2, b-1, c-3, d-3 D) a-3, b-2, c-1, d-2

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Answer & Solution

Answer: Option A

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37

If a and b are constants, the most general solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 2\frac{{{\text{dx}}}}{{{\text{dt}}}} + {\text{x}} = 0$$ is

A) ae^-t B) ae^-t + bte^-t C) ae^t + bte^-t D) ae^-2t

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Answer & Solution

Answer: Option B

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38

The solution of the differential equation, for t > 0, y''(t) + 2y'(t) + y(t) = 0 with initial conditions y(0) = 0 and y'(0) = 1, is (u(t) denotes the unit step function),

A) te^-tu(t) B) (e^-t - te^-t)u(t) C) (-e^-t + te^-t)u(t) D) e^-tu(t)

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Answer & Solution

Answer: Option A

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39

A differential equation is given as
$${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 2{\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{y}} = 4$$
The solution of differential equation in terms of arbitrary constant C₁ and C₂ is

A) $${\text{y}} = \frac{{{{\text{C}}_1}}}{{{{\text{x}}^2}}} + {{\text{C}}_2}{\text{x}} + 2$$ B) $${\text{y}} = {{\text{C}}_1}{{\text{x}}^2} + {{\text{C}}_2}{\text{x}} + 4$$ C) $${\text{y}} = {{\text{C}}_1}{{\text{x}}^2} + {{\text{C}}_2}{\text{x}} + 2$$ D) $${\text{y}} = \frac{{{{\text{C}}_1}}}{{{{\text{x}}^2}}} + {{\text{C}}_2}{\text{x}} + 4$$

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Answer & Solution

Answer: Option C

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40

The solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 0$$ with boundary conditions
$${\text{i}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 0;\,{\text{ii}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 1{\text{ is}}$$

A) y = 1 B) y = x C) y = x + c where C is an arbitrary constant D) y = C₁x + C₂ where C₁ and C₂ are arbitrary constants

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Answer & Solution

Answer: Option C

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