partial differential equations

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

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 :...

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)...

A 5 cm long metal rod AB was initially at a uniform temperature of T₀ °C. Thereafter, temperature at both the ends are maintained at 0 °C. Neglecting the heat transfer from the lat...

A 5 cm long metal rod AB was initially at a uniform temperature of T 0 °C. Thereafter, temperature at both the ends are maintained at 0°C. Neglecting the heat transfer from the lat...

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 $x = 0, y = 0$. Th...

Consider the following expression: z = sin(y + it) + cos(y $$-$$ it) where z, y, and t are variables, and $$i = \sqrt { - 1} $$ is a complex number. The partial differential equati...

The solution of the partial differential equation $${{\partial u} \over {\partial t}} = \alpha {{{\partial ^2}u} \over {\partial {x^2}}}$$ is of the form

The partial differential equation that can be formed from $$z=ax+by+ab$$ has the form $$\,\,\left( {p = {{\partial z} \over {\partial x}},q = {{\partial z} \over {\partial y}}} \ri...

The number of boundary conditions required to solve the differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} =...