1
GATE ECE 2024
+1
-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 ____.

A

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

B

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

C

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

D

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

2
GATE ECE 2022
MCQ (More than One Correct Answer)
+1
-0.33

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.

A
$$\xi = {1 \over {\sqrt {2a} }},\eta {1 \over {\sqrt {2b} }}$$
B
$$\xi = {1 \over {\sqrt a }},\eta = 0$$
C
$$\xi = 0,\,\eta = 0$$
D
$$\xi = {1 \over {\sqrt a }},\eta {1 \over {\sqrt b }}$$
3
GATE ECE 2017 Set 2
+1
-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
A
$${K_1}{e^{\left( { - 1 + \sqrt 6 } \right)x}} + {K_2}{e^{\left( { - 1 - \sqrt 6 } \right)x}}$$
B
$${K_1}{e^{\left( { - 1 + \sqrt 8 } \right)x}} + {K_2}{e^{\left( { - 1 - \sqrt 8 } \right)x}}$$
C
$${K_1}{e^{\left( { - 2 + \sqrt 6 } \right)x}} + {K_2}{e^{\left( { - 2 - \sqrt 6 } \right)x}}$$
D
$${K_1}{e^{\left( { - 2 + \sqrt 8 } \right)x}} + {K_2}{e^{\left( { - 2 - \sqrt 8 } \right)x}}$$
4
GATE ECE 2017 Set 1
+1
-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?

A
Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ are true
B
Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$ are true
C
Both $${\rm I}$$$${\rm I}$$ and $${\rm IV}$$ are true
D
Both $${\rm I}$$$${\rm I}$$$${\rm I}$$ and $${\rm IV}$$ are true
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