Differential Equations MCQs Question and Answer in Engineering Maths

71

The degree of the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 2{{\text{x}}^3} = 0$$ is

A) 0 B) 1 C) 2 D) 3

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Answer & Solution

Answer: Option B

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72

The solution for the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} = - 9{\text{x}}$$ with initial conditions x(0) = 1 and $${\left. {\frac{{{\text{dx}}}}{{{\text{dt}}}}} \right|_{{\text{t}} = 0}} = 1,$$ is

A) t² + t + 1 B) $$\sin 3{\text{t}} + \frac{1}{3}\cos 3{\text{t}} + \frac{2}{3}$$ C) $$\frac{1}{3}\sin 3{\text{t}} + \cos 3{\text{t}}$$ D) cos 3t + t

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Answer: Option C

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73

Consider the differential equation: $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left( {1 + {{\text{y}}^2}} \right){\text{x}}{\text{.}}$$
The general solution with constant c is

A) $${\text{y}} = \tan \frac{{\text{x}}}{2} + \tan {\text{c}}$$ B) $${\text{y}} = {\tan ^2}\left( {\frac{{\text{x}}}{2} + {\text{c}}} \right)$$ C) $${\text{y}} = {\tan ^2}\left( {\frac{{\text{x}}}{2}} \right) + {\text{c}}$$ D) $${\text{y}} = \tan \left( {\frac{{{{\text{x}}^2}}}{2} + {\text{c}}} \right)$$

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Answer: Option D

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74

A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in

A) 6 months B) 9 months C) 12 months D) infinite time

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Answer: Option A

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75

The particular solution of the initial value problem given below is
$$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 12\frac{{{\text{dy}}}}{{{\text{dx}}}} + 36{\text{y}} = 0$$ with y(0) = 3 and $${\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = 0}} = - 36$$

A) (3 - 18x)e^-6x B) (3 + 25x)e^-6x C) (3 + 20x)e^-6x D) (3 - 12x)e^-6x

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Answer & Solution

Answer: Option A

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76

The solution to x²y'' + y' - y = 0 is

A) y = c₁x² + c₂x^-3 B) y = c₁ + c₂x^-2 C) $${\text{y}} = {{\text{c}}_1}{\text{x}} + \frac{{{{\text{c}}_2}}}{{\text{x}}}$$ D) y = c₁x + c₂x⁴

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Answer: Option C

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77

Consider the following difference equation
$${\text{x}}\left( {{\text{ydx}} + {\text{xdy}}} \right)\cos \frac{{\text{y}}}{{\text{x}}} = {\text{y}}\left( {{\text{xdy}} - {\text{ydx}}} \right)\sin \frac{{\text{y}}}{{\text{x}}}$$
Which of the following is the solution of the above equation (c is an arbitrary constant)?

A) $$\frac{{\text{x}}}{{\text{y}}}\cos \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$ B) $$\frac{{\text{x}}}{{\text{y}}}\sin \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$ C) $${\text{xy}}\cos \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$ D) $${\text{xy}}\sin \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$

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Answer: Option C

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78

Consider the differential equation $${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} - {\text{y}} = 0.$$ Which of the following is a solution to this differential equation for x > 0 ?

A) e^x B) x² C) $$\frac{1}{{\text{x}}}$$ D) $$l$$n x

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Answer & Solution

Answer: Option C

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79

The complete integral of (z - px - qy)³ = pq + 2(p² + q)² is

A) $${\text{z}} = {\text{ax}} + {\text{by}} + \root 3 \of {{\text{pq}} + 2{{\left( {{{\text{p}}^2} + {\text{q}}} \right)}^2}} $$ B) $${\text{z}} = {\text{ax}} + {\text{by}} + \root 3 \of {{\text{ab}} + 2{{\left( {{{\text{a}}^2} + {\text{b}}} \right)}^2}} $$ C) $${\text{z}} = {\text{ax}} + {\text{by}} + \root 3 \of {{\text{ab}}} + \root 3 \of {2{{\left( {{{\text{a}}^2} + {\text{b}}} \right)}^2}} $$ D) $${\text{z}} = {\text{ax}} + {\text{by}} + {\text{c}}$$

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Answer: Option B

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80

The integrating factor for differential equation $$\frac{{{\text{dP}}}}{{{\text{dt}}}} + {{\text{k}}_2}{\text{P}} = {{\text{k}}_1}{{\text{L}}_0}{{\text{e}}^{ - {{\text{k}}_1}{\text{t}}}}$$ is

A) e^-k₁t B) e^-k₂t C) e^k₁t D) e^k₂t

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Answer: Option D

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