Differential Equations MCQs Question and Answer in Engineering Maths

81

Which one of the following differential equation has a solution given by function $${\text{y}} = 5\sin \left( {3{\text{x}} + \frac{\pi }{3}} \right)$$

A) $$\frac{{{\text{dy}}}}{{{\text{dx}}}} - \frac{5}{3}\cos \left( {3{\text{x}}} \right) = 0$$ B) $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + \frac{5}{3}\cos \left( {3{\text{x}}} \right) = 0$$ C) $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 9{\text{y}} = 0$$ D) $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 9{\text{y}} = 0$$

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Answer: Option C

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82

The Blasius equation, $$\frac{{{{\text{d}}^3}{\text{f}}}}{{{\text{d}}{{\text{n}}^3}}} + \frac{{\text{f}}}{2}\frac{{{{\text{d}}^2}{\text{f}}}}{{{\text{d}}{{\text{n}}^2}}} = 0,$$ is a

A) second order nonlinear ordinary differential equation B) third order nonlinear ordinary differential equation C) third order linear ordinary differential equation D) mixed order nonlinear ordinary differential equation

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Answer: Option B

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83

If the characteristic equation of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 2\alpha \frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{y}} = 0$$ has two equal roots, then the values of $$\alpha $$ are

A) ±1 B) 0, 0 C) ±j D) $$ \pm \frac{1}{2}$$

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Answer: Option A

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84

The solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + \frac{{\text{y}}}{{\text{x}}} = {\text{x}},$$ with the condition that y = 1 at x = 1, is

A) $${\text{y}} = \frac{2}{{3{{\text{x}}^2}}} + \frac{{\text{x}}}{3}$$ B) $${\text{y}} = \frac{{\text{x}}}{2} + \frac{{\text{1}}}{{2{\text{x}}}}$$ C) $${\text{y}} = \frac{2}{3} + \frac{{\text{x}}}{3}$$ D) $${\text{y}} = \frac{2}{{3{\text{x}}}} + \frac{{{{\text{x}}^2}}}{3}$$

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Answer: Option D

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85

Consider the following differential equation: $$\frac{{{\text{dy}}}}{{{\text{dt}}}} = - 5{\text{y;}}$$ initial condition: y = 2 at t = 0. The value of y at t = 3 is

A) -5e^-10 B) 2e^-10 C) 2e^-15 D) -15e²

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Answer: Option C

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86

Which of the following is a solution to the differential equation $$\frac{{{\text{dx}}\left( {\text{t}} \right)}}{{{\text{dt}}}} + 3{\text{x}}\left( {\text{t}} \right) = 0?$$

A) x(t) = 3e^-t B) x(t) = 2e^-3t C) x(t) = $$ - \frac{3}{2}$$ t² D) x(t) = 3t²

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Answer: Option B

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87

The homogenous part of differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{P}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{qy}} = {\text{r}}$$ (P, q, r are constants) has real distinct roots if

A) P² - 4q > 0 B) P² - 4q < 0 C) P² - 4q = 0 D) P² - 4q = r

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Answer: Option A

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88

For $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 4\frac{{{\text{dy}}}}{{{\text{dx}}}} + 3{\text{y}} = 3{{\text{e}}^{2{\text{x}}}},$$ the particular integral is

A) $$\frac{1}{{15}}{{\text{e}}^{2{\text{x}}}}$$ B) $$\frac{1}{5}{{\text{e}}^{2{\text{x}}}}$$ C) 3e^2x D) C₁e^-x + C₂e^-3x

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Answer: Option B

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89

The solution of $${\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{y}} = {{\text{x}}^4}$$ with the condition $${\text{y}}\left( 1 \right) = \frac{6}{5}$$ is

A) $${\text{y}} = \frac{{{{\text{x}}^4}}}{5} + \frac{1}{{\text{x}}}$$ B) $${\text{y}} = \frac{{4{{\text{x}}^4}}}{5} + \frac{4}{{5{\text{x}}}}$$ C) $${\text{y}} = \frac{{{{\text{x}}^4}}}{5} + 1$$ D) $${\text{y}} = \frac{{{{\text{x}}^5}}}{5} + 1$$

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Answer: Option A

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90

The solution of differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + \frac{{{\text{6dy}}}}{{{\text{dx}}}} + 9{\text{y}} = 9{\text{x}} + 6$$ with C₁ and C₂ as constant is

A) y = [C₁x + C₂]e^-3x B) y = C₁e^3x + C₂e^-3x C) y = (C₁x + C₂)e^-3x + x D) y = (C₁x + C₂)e^3x + x

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Answer & Solution

Answer: Option C

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