HPAS Maths Optional PYQs: Partial Differential Equations

On that page, you will find year-wise and question-wise solutions.

Based on your request and drawing solely on the information provided in the sources (HPAS Maths Optional Question Papers 2014–2024), here is the compilation of all questions related to Partial Differential Equations (PDE).

HPAS 2024 Partial Differential Equations Questions

Question 5(a): Solve the partial differential equation (Lagrange’s method type): $$ \mathbf{(x^2 – yz)p + (y^2 – zx)q = z^2 – xy} $$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.

Question 5(b): Using Monge’s method, solve the wave equation $\mathbf{r = a^2t}$ (for $a>0$), where $r = \frac{\partial^2 z}{\partial x^2}$ and $t = \frac{\partial^2 z}{\partial y^2}$.

HPAS 2023 Partial Differential Equations Questions

Question 6(a): Solve the partial differential equation (Lagrange’s method type): $$ \mathbf{\left(\frac{y-z}{yz}\right)p + \left(\frac{z-x}{zx}\right)q = \frac{x-y}{xy}} $$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.

Question 6(b): Using Charpit’s method, find the solution of the partial differential equation $\mathbf{p^2x + q^2y = z}$, where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.

Question 6(c): Solve the partial differential equation (Linear PDE with constant coefficients): $\mathbf{(D^2 – DD’ + D’ – 1)z = \cos(x+2y) + e^y}$, where $D = \frac{\partial}{\partial x}$ and $D’ = \frac{\partial}{\partial y}$.

HPAS 2021 Partial Differential Equations Questions

Question 5(a): Solve the following partial differential equation using the Lagrange method: $\mathbf{p(z+e^x) + q(z+e^y) = z^2 – e^{x+y}}$.

Question 6(a): Solve the partial differential equation (Linear Homogeneous PDE): $\mathbf{(D^2 + DD’ – 6D’^2)z = y \cos x}$, where $D = \frac{\partial}{\partial x}$ and $D’ = \frac{\partial}{\partial y}$.

Question 6(b): Solve the partial differential equation using Monge’s method: $\mathbf{r – t \sin^2 x – p \cot x = 0}$.

Question 6(c): Find the complete integral of the partial differential equation: $\mathbf{2\sqrt{p} + 3\sqrt{q} = 6x + 2y}$.

HPAS 2020 Partial Differential Equations Questions

Question 1(c): Solve (Lagrange’s method type): $\mathbf{x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = -xy}$.

Question 7(a): Find the general solution of the partial differential equation (Linear PDE with constant coefficients): $\mathbf{D(D^2 – D’ + 1)(D + 2D’ + 1)^3 z = 0}$.

Question 7(b): Apply Charpit’s method to find the complete integral of the partial differential equation $\mathbf{xp^2 + yq^2 = z}$.

HPAS 2019 Partial Differential Equations Questions

Question 7(b): Solve the partial differential equation (Lagrange’s method type): $\mathbf{(x^2 – y^2 – z^2)p + 2xyq = 2xz}$.

HPAS 2018 Partial Differential Equations Questions

Question 4(a): Obtain the partial differential equation for the set of all right circular cones whose axes coincide with the z-axis.

Question 4(b): Prove that any sufficiently differentiable function of the form $\mathbf{F(x+kt)}$ satisfies the wave equation $\mathbf{F_{xx} = (1/k^2)F_{tt}}$.

HPAS 2017 Partial Differential Equations Questions

Question 6(a): Solve the following equation by Charpit’s method: $\mathbf{(p^2+q^2)y = qz}$.

Question 6(b): Solve (Homogeneous Linear PDE): $\mathbf{x^2r + 2xys + y^2t = 0}$.

HPAS 2016 Partial Differential Equations Questions

Question 1(e): Form a partial differential equation by eliminating the arbitrary function from the equation: $\mathbf{lx+my+nz=\phi(x^{2}+y^{2}+z^{2})}$.

Question 8(b): Find a complete integral of the partial differential equation $\mathbf{(p + q) (px+ qy) = 1}$.

HPAS 2015 Partial Differential Equations Questions

Question 1(d): Form a partial differential equation by eliminating the functions from the equation: $\mathbf{Z=f(x+iy)+\phi(x-iy)}$.

Question 6(a): Solve (Lagrange’s method type): $\mathbf{(y+z)p+(z+x)q=(x+y)}$.

Question 6(b): Solve (Linear Non-Homogeneous PDE): $\mathbf{r+s-6t=y~\cos~x}$.

HPAS 2014 Partial Differential Equations Questions

Question 1(e): Solve the following partial differential equation: $\mathbf{p^{2}+q^{2}-2px-2qy+1=0}$.

Question 6(a): Solve (Non-linear PDE): $\mathbf{pxy+pq+qy=yz}$.

Question 6(b): Solve by Monge’s method: $\mathbf{pq=x(ps-qr)}$.