Partial Differential Equation (PDE) - Engineering Mathematics - Civil Engineering Previous Year Questions

1

2016 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2016 [Session 1]

The type of partial differential equation \(\frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} + 3 \frac{\partial^2 p}{\partial x \partial y} + 2 \frac{\partial p}{\partial x} - \frac{\partial p}{\partial y} = 0\) is

A

elliptic

B

parabolic

C

hyperbolic

D

none of these

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2

2016 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2016 [Session 1]

The solution of the partial differential equation \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\) is of the form

A

\(C \cos(kt) [C_1 e^{\sqrt{k/\alpha}x} + C_2 e^{-\sqrt{k/\alpha}x}]\)

B

\(C e^{kt} [C_1 e^{\sqrt{k/\alpha}x} + C_2 e^{-\sqrt{k/\alpha}x}]\)

C

\(C e^{kt} [C_1 \cos(\sqrt{k/\alpha}x) + C_2 \sin(-\sqrt{k/\alpha}x)]\)

D

\(C \sin(kt) [C_1 \cos(\sqrt{k/\alpha}x) + C_2 \sin(-\sqrt{k/\alpha}x)]\)

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3

2020 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2020 [Session 2]

The following partial differential equation is defined for \(u: u(x,y)\): \( \frac{\partial u}{\partial y} = \frac{\partial^2 u}{\partial x^2} ; \quad y \ge 0; \quad x_1 \le x \le x_2 \)
The set of auxiliary conditions necessary to solve the equation uniquely, is

A

three initial conditions

B

three boundary conditions

C

two initial conditions and one boundary condition

D

one initial condition and two boundary conditions

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4

2022 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2022 [Session 2]

The function \( f(x, y) \) satisfies the Laplace equation
\[ \nabla^2 f(x, y) = 0 \]
on a circular domain of radius \( r = 1 \) with its center at point P with coordinates \( x = 0, y = 0 \). The value of this function on the circular boundary of this domain is equal to 3.
The numerical value of \( f(0, 0) \) is:

A

0

B

2

C

3

D

1

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5

2024 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2024 [Session 1]

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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6

2024 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2024 [Session 1]

For the following partial differential equation, \( x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2} \) which of the following option(s) is/are CORRECT?

A

elliptic for \(x > 0 \) and \(y > 0 \)

B

parabolic for \(x > 0 \) and \(y > 0 \)

C

elliptic for \(x = 0 \) and \(y > 0 \)

D

hyperbolic for \(x < 0 \) and \(y > 0 \)

Check answer

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7

2024 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2024 [Session 2]

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 \(2\text{ m} \times 2\text{ m}\). 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

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8

2025 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2025 [Session 1]

Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:

A

\(\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + xy\)

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

\(\left(\frac{\partial^2 u}{\partial t^2}\right)^2 = c^2 \frac{\partial^2 u}{\partial x^2}\)

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9

2010 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2010

The partial differential equation that can be formed from \( z = ax + by + ab \) has the form (with \( p = \frac{\partial z}{\partial x} \) and \( q = \frac{\partial z}{\partial y} \))

A

\(z = px + qy \)

B

\(z = px + qy + pq \)

C

\(z = px + qy + p q \)

D

\(z = qy + pq \)

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10

2026 · Civil Engineering · Engineering Mathematics · Partial Differential Equation (PDE)

Civil Engineering (CE) 2026

A partial differential equation is given below.
\[\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0\]
Possible solution(s) is/are:

A

\((x + y)^5\)

B

\((x - 2y)^3\)

C

\(\cos(x + y)\)

D

\(\sin(x - 2y)\)

Check answer

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