List of top Engineering Mathematics Questions on Partial Differential Equations

Consider the ordinary differential equation: \[ \frac{1}{2}\frac{dy}{dx} + \frac{y}{x} = 1. \] If \( y = \frac{2}{3} \) at \( x = 1 \), then the value of \( y \) at \( x = 3 \) is ___________ (rounded off to the nearest integer).
For all real values of \( x \) and \( y \), the partial differential equation in terms of \( \psi(x, y) \) given by \[ \frac{\partial^2 \psi}{\partial x^2} + 2 \frac{\partial^2 \psi}{\partial x \partial y} - 3 \frac{\partial^2 \psi}{\partial y^2} = 0 \] is ________

Consider the ordinary differential equation:

The partial differential equation \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \] is ________.

Consider the ordinary differential equation: \[ \frac{1}{2} \frac{dy}{dx} + \frac{y}{x} = 1. { If } y = \frac{2}{3} { at } x = 1, { then the value of } y { at } x = 3 { is } \_\_\_\_\_\_ { (rounded off to the nearest integer).} \]
The partial differential equation \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \] is \_\_\_\_\_\_.
Consider the ordinary differential equation: \[ \frac{1}{2} \frac{dy}{dx} + \frac{y}{x} = 1. { If } y = \frac{2}{3} { at } x = 1, { then the value of } y { at } x = 3 { is } \_\_\_\_\_\_ { (rounded off to the nearest integer).} \]
The partial differential equation \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \] is \_\_\_\_\_\_.
Consider the ordinary differential equation: \[ \frac{1}{2} \frac{dy}{dx} + \frac{y}{x} = 1. { If } y = \frac{2}{3} { at } x = 1, { then the value of } y { at } x = 3 { is } \_\_\_\_\_\_ { (rounded off to the nearest integer).} \]
The partial differential equation obtained by eliminating the constants \(a\) and \(b\) from \[ z = (x - a)^2 + (y - b)^2 \] is:
If the inverse Laplace transform of \( e^{-s} \cot^{-1}s \) is \( f(t) \), then \( f\left( \frac{3\pi}{2} \right) = \)
If the function \( u = \sin^{-1} \left( \frac{x^3 + y^3}{x + y} \right) \), then \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \] is
Let $ u(x,t) $ satisfy the initial boundary value problem $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad x \in (0,1), t>0 $$ given that $$ u(x, 0) = \sin(\pi x), \quad x \in [0,1], \quad u(0,t) = u(1,t) = 0, \quad t>0. $$ Then $ u \left( x, \frac{1}{\pi^2} \right) $ for $ x \in (0,1) $ is
Complete integral of the partial differential equation \( 2\frac{\partial^2 f}{\partial q^2} + 3\frac{\partial f}{\partial q} = 6x + 2y \) is
The solution of the given ordinary differential equation \( x \frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0 \) is:
The complete integral of the partial differential equation \( pz^2 \sin^2 x + qz^2 \cos^2 y = 1 \) is:
The complete integral of the partial differential equation \( pz^2 \sin^2 x + qz^2 \cos^2 y = 1 \) is:
Consider the one–dimensional wave (advection) equation \[ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=0, -\infty<x<\infty,\ t\ge 0. \] For the initial condition \(u(x,0)=e^{-x^{2}}\), the solution at \(t=1\) is:
The real function \( y = \sin^2(|x|) \) is
Consider the differential equation \(\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0\). The boundary conditions are \(y = 0\) and \(\frac{dy}{dx} = 1\) at \(x = 0\). Then the value of \(y\) at \(x = \frac{1}{2}\) is
Consider the partial differential equation \(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0\) where \(x, y\) are real. If \(f(x, y) = a(x)b(y)\), where \(a(x)\) and \(b(y)\) are real functions, which one of the following statements can be true?
The arc length of the parametric curve: \( x=\cos\theta,\; y=\sin\theta,\; z=\theta \) from \( \theta=0 \) to \( \theta=2\pi \) is equal to _________ (round off to one decimal place).