1
GATE CE 2025 Set 1
MCQ (Single Correct Answer)
+1
-0.33
Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:
A
$\frac{\partial^2 u}{\partial t^2}=c^2\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)+x y$
B
$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}=0$
C
$\frac{\partial u}{\partial t}=c \frac{\partial u}{\partial x}$
D
$\left(\frac{\partial^2 u}{\partial t^2}\right)^2=c^2 \frac{\partial^2 u}{\partial x^2}$
2
GATE CE 2024 Set 2
MCQ (Single Correct Answer)
+1
-0.33

A partial differential equation

$$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$

is defined for the two-dimensional field $T: T(x, y)$, inside a planar square domain of size 2 m × 2 m. Three boundary edges of the square domain are maintained at value $T = 50$, whereas the fourth boundary edge is maintained at $T = 100$.

The value of $T$ at the center of the domain is

A

50.0

B

62.5

C

75.0

D

87.5

3
GATE CE 2024 Set 2
MCQ (Single Correct Answer)
+1
-0.33

Consider two Ordinary Differential Equations (ODEs):

P: $ \dfrac{dy}{dx} = \dfrac{x^4 + 3x^2 y^2 + 2y^4}{x^3 y} $

Q: $ \dfrac{dy}{dx} = -\dfrac{y^2}{x^2} $

Which one of the following options is CORRECT?

A

P is a homogeneous ODE and Q is an exact ODE.

B

P is a homogeneous ODE and Q is not an exact ODE.

C

P is a nonhomogeneous ODE and Q is an exact ODE.

D

P is a nonhomogeneous ODE and Q is not an exact ODE.

4
GATE CE 2024 Set 1
MCQ (Single Correct Answer)
+1
-0.33

The second-order differential equation in an unknown function $$u : u(x, y)$$ is defined as $$\frac{\partial^2 u}{\partial x^2}= 2$$

Assuming $$g : g(x)$$, $$f : f(y)$$, and $$h : h(y)$$, the general solution of the above differential equation is

A

$$u = x^2 + f(y) + g(x)$$

B

$$u = x^2 + x f(y) + h(y)$$

C

$$u = x^2 + x f(y) + g(x)$$

D

$$u = x^2 + f(y) + y g(x)$$

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