PYQs Based Test - 15 : Partial Differential Equation

The solution of the partial differential equation $x^2\frac{\partial z}{\partial x}+y^2\frac{\partial z}{\partial y}=\left(x+y\right)z$ is

The partial differential equation $\frac{{\partial }^{2}u}{\partial t^2}-c^2\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)=0$; where c ≠ 0 is known as

The solution at x = 1, t = 1 of the partial differential equation, $\frac{{\partial }^{2}u}{\partial x^2}=25\frac{{\partial }^{2}u}{d{t}^2}$ subject to initial condition of $u\left(0\right)=3x,\frac{\partial u}{\partial t}\left(0\right)=3$ is _____.

The following partial differential equation is defined for u:u (x, y) $\frac{\partial u}{\partial y}=\frac{{\partial }^{2}u}{\partial x^2};y\ge 0;x_1\le x\le x_2$

The set auxiliary conditions necessary to solve the equation uniquely, is

Consider the following partial differential equation for with the constant c > 1:

Solution of this equation is $\frac{\partial u}{\partial y}+c\frac{\partial u}{\partial x}=0$