GATE ECE 2024 | Differential Equations Question 1 | Engineering Mathematics

GATE ECE 2024

MCQ (Single Correct Answer)

-0.33

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is ____.

$\dfrac{d^2 y}{d t^2} + 4 \dfrac{d y}{d t} + 4 y = f(t)$

$\dfrac{d^2 y}{d t^2} + 4 y = f(t)$

$\dfrac{d^2 y}{d t^2} + 3 \dfrac{d y}{d t} + 2 y = f(t)$

$\dfrac{d^2 y}{d t^2} + 5 \dfrac{d y}{d t} + 6 y = f(t)$

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 2017 Set 1

MCQ (Single Correct Answer)

-0.3

Consider the following statements about the linear dependence of the real valued functions $${y_1} = 1,\,\,{y_2} = x$$ and $${y_3} = {x^2}$$. Over the field of real numbers.

$${\rm I}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly independent on $$ - 1 \le x \le 0$$
$${\rm II}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly dependent on $$0 \le x \le 1$$
$${\rm III}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly independent on $$0 \le x \le 1$$
$${\rm IV}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly dependent on $$ - 1 \le x \le 0$$

Which one among the following is correct?

Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ are true

Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$ are true

Both $${\rm I}$$$${\rm I}$$ and $${\rm IV}$$ are true

Both $${\rm I}$$$${\rm I}$$$${\rm I}$$ and $${\rm IV}$$ are true

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

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

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

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

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