GATE ECE 2022 | Differential Equations Question 4 | Engineering Mathematics

GATE ECE 2022

MCQ (More than One Correct Answer)

-0

Consider the following partial differential equation (PDE)

$$a{{{\partial ^2}f(x,y)} \over {\partial {x^2}}} + b{{{\partial ^2}f(x,y)} \over {\partial {y^2}}} = f(x,y)$$,

where a and b are distinct positive real numbers. Select the combination(s) of values of the real parameters $$\xi $$ and $$\eta $$ such that $$f(x,y) = {e^{\xi x + \eta y}}$$ is a solution of the given PDE.

$$\xi = {1 \over {\sqrt {2a} }},\eta {1 \over {\sqrt {2b} }}$$

$$\xi = {1 \over {\sqrt a }},\eta = 0$$

$$\xi = 0,\,\eta = 0$$

$$\xi = {1 \over {\sqrt a }},\eta {1 \over {\sqrt b }}$$

GATE ECE 2021

MCQ (Single Correct Answer)

-0.33

Consider the differential equation given below. $${{dy} \over {dx}} + {x \over {1 - {x^2}}}y = x\sqrt y $$

The integrating factor of the differential equation is

$${(1 - {x^2})^{ - {1 \over 2}}}$$

$${(1 - {x^2})^{ - {3 \over 4}}}$$

$${(1 - {x^2})^{ - {3 \over 2}}}$$

$${(1 - {x^2})^{ - {1 \over 4}}}$$

GATE ECE 2020

MCQ (Single Correct Answer)

-0.33

The general solution of $\frac{d^2 y}{d x^2}-6 \frac{d y}{d x}+9 y=0$ is

$y=C_1 e^{3 x}+C_2 e^{-3 x}$

$y=C_1 e^{3 x}$

$y=\left(C_1+C_2 x\right) e^{3 x}$

$y=\left(C_1+C_2 x\right) e^{-3 x}$

GATE ECE 2017 Set 2

MCQ (Single Correct Answer)

-0.3

The general solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} + 2{{dy} \over {dx}} - 5y = 0\,\,\,$$ in terms of arbitrary constants $${K_1}$$ and $${K_2}$$ is

$${K_1}{e^{\left( { - 1 + \sqrt 6 } \right)x}} + {K_2}{e^{\left( { - 1 - \sqrt 6 } \right)x}}$$

$${K_1}{e^{\left( { - 1 + \sqrt 8 } \right)x}} + {K_2}{e^{\left( { - 1 - \sqrt 8 } \right)x}}$$

$${K_1}{e^{\left( { - 2 + \sqrt 6 } \right)x}} + {K_2}{e^{\left( { - 2 - \sqrt 6 } \right)x}}$$

$${K_1}{e^{\left( { - 2 + \sqrt 8 } \right)x}} + {K_2}{e^{\left( { - 2 - \sqrt 8 } \right)x}}$$

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