Differential Equations MCQs Question and Answer in Engineering Maths Page-8 section-1

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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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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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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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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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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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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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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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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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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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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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

Answer & Solution

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

Answer & Solution

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

Answer & Solution

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

Answer & Solution

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 $${\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = 0}} = - 36$$

Answer & Solution

76. The solution to x2y'' + y' - y = 0 is

Answer & Solution

Answer & Solution

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 ?

Answer & Solution

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

Answer & Solution

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

Answer & Solution

71.
The degree of the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 2{{\text{x}}^3} = 0$$ is

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

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

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

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 $${\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = 0}} = - 36$$

76.
The solution to x²y'' + y' - y = 0 is

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 ?

79.
The complete integral of (z - px - qy)³ = pq + 2(p² + q)² is

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