GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Old Question Papers PDF

Express the complex number $(1 + \sqrt{3}i)^2$ in polar form.

Summer 2026 Q1a · Summer 2024 Q1a · Winter 2023 Q1a · Summer 2023 Q3b OR

Form partial differential equation from $z = ax + by + a$ by eliminating the arbitrary constant $a$ and $b$.

Summer 2026 Q4a OR · Winter 2024 Q4a · Winter 2023 Q4a OR · Summer 2023 Q4b

Using the method of separation of variables, solve $u_x = 2u_t + u$ with $u(x, 0) = 6e^{-3x}$.

Summer 2026 Q5a OR · Winter 2025 Q5b · Winter 2024 Q5c OR · Winter 2023 Q5c OR

Show that $u(x, y) = 2x - x^3 + 3xy^2$ is harmonic and find its harmonic conjugate $v(x, y)$.

Summer 2026 Q1c · Summer 2025 Q2c OR · Summer 2023 Q1c

Determine an analytic function with real part $u(x, y) = \frac{y}{x^2 + y^2}$.

Summer 2026 Q2c OR · Summer 2025 Q1c · Summer 2023 Q1a

Evaluate following real integration using residue theorem: $\int_{-\infty}^\infty \frac{\cos x}{x^2 + 1} \, dx$.

Winter 2025 Q3c OR · Winter 2024 Q3c OR · Winter 2023 Q5a

Solve Laplace's equation in square lamina with $u(0, y) = u(\pi, y) = u(x, \pi) = 0$ and $u(x, 0) = T_0$, constant.

Summer 2026 Q5c OR · Summer 2025 Q5c OR · Summer 2024 Q5c

Show that $u(x,y) = x^2 - y^2 + x$ is harmonic. Find the corresponding analytic function $f(z) = u + iv$.

Winter 2024 Q3b · Summer 2024 Q1b · Winter 2023 Q2c

Find the radii of convergence and region of convergence of the following: (i) $\sum_{n=1}^\infty \frac{z^n}{2^{n+1}}$ (ii) $\sum_{n=1}^\infty \frac{n!}{n^n} z^n$

Winter 2025 Q3b · Summer 2024 Q2b · Winter 2023 Q3a

Find Maclaurin series expansion of $f(z) = \frac{1}{z + 3i}$ and its radius of convergence.

Summer 2026 Q3b OR · Winter 2025 Q3b OR · Winter 2023 Q3b OR

Solve $p^2 + q^2 = x + y$.

Summer 2025 Q5b · Winter 2024 Q5a · Winter 2023 Q4b

Solve $\frac{\partial^2 z}{\partial x^2} + 4 \frac{\partial^2 z}{\partial x \partial y} - 5 \frac{\partial^2 z}{\partial y^2} = \sin(2x + 3y)$.

Summer 2025 Q4b OR · Winter 2024 Q5c · Winter 2023 Q1c

Solve the second order homogeneous partial differential equation: $z_{xx} - 2z_{xy} - 8z_{yy} = 0$.

Summer 2026 Q4b OR · Winter 2024 Q5b OR · Winter 2023 Q5a OR

Classify the second order homogeneous partial differential equation $\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} - 12\frac{\partial^2 z}{\partial y^2} = 0$ as elliptic, parabolic or hyperbolic.

Summer 2026 Q4b · Winter 2025 Q5a OR · Summer 2024 Q5a

Discuss about the types of singularities of $f(z)$.

Summer 2026 Q2a · Winter 2025 Q3a OR · Summer 2024 Q3a

Classify the poles of $f(z) = \frac{1}{z^2 - z^6}$.

Summer 2025 Q3a OR · Summer 2024 Q3a OR · Winter 2023 Q4a

Find all values of $(-8i)^{1/3}$ using De Moivre's theorem.

Summer 2026 Q3a · Winter 2024 Q1b · Winter 2023 Q1b

Using Residue theorem evaluate the integral $\int_0^\pi \frac{1}{17 - 8 \cos \theta} \, d\theta$.

Summer 2025 Q3c OR · Summer 2024 Q3c OR

Evaluate $\int_C \text{Re}(z^2) \, dz$, where $C$ is the boundary of the square with vertices $0, i, 1 + i, 1$ in the clockwise wise direction.

Summer 2025 Q3c · Summer 2023 Q2a

Explain inversion transformation. Find the image of $\frac{1}{4} \le y \le \frac{1}{2}$ under mapping $w = 1/z$. Show this translation on graph.

Winter 2025 Q2c · Winter 2024 Q2c OR

Derive the solutions of one dimensional wave equation.

Winter 2025 Q5c OR · Summer 2024 Q5c OR

Find the solution of the wave equation $\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}$ such that $y = a \cos pt$ when $x = l$, and $y = 0$ when $x = 0$.

Winter 2023 Q5c · Summer 2023 Q5c OR

Solve following non-linear partial differential equations: a) $p(1 + q) = qz$ b) $p^2 - q^2 = x - y$

Winter 2025 Q4b OR · Summer 2024 Q4b OR

Solve $\frac{\partial^3 z}{\partial x^2 \partial y} = \sin(3x + 2y)$.

Summer 2025 Q4b · Summer 2025 Q4c OR · Summer 2024 Q5a OR

Using method of separation of variables solve $\frac{\partial u}{\partial t} + u = 2\frac{\partial u}{\partial x}$ given that $u(x, 0) = 4e^{-3x}$.

Summer 2024 Q5b · Summer 2023 Q5b OR

Evaluate the integral $\int_C \bar{z} \, dz$, where $C$ is right half of the circle $z = 2e^{i\theta} \left(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\right)$.

Winter 2025 Q2a · Summer 2024 Q2a

Solve the partial differential equation using the method of separation of variables: $u_x + 2u_y = 0$, given that $u(x, 0) = 4e^{-x}$.

Summer 2026 Q5a · Summer 2025 Q5c