List of top Mathematics Questions on Partial Differential Equations

If $(x/a)^n + (y/b)^n = 2$, then $dy/dx$ at $(a,b)$ is

The solution of $\frac{\partial^2 z}{\partial x^2} + z = 0$, satisfying $z(0,y) = e^y$, $\left(\frac{\partial z}{\partial x}\right)_{x=0} = 1$ is $z(x,y) = $}

If $z = x^2 y + e^{xy^2}$, then $\left(\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial x \partial y}\right)$ evaluated at $(1,0)$ is:}

The differential equation \[ 2\frac{\partial^2 z}{\partial x^2} + 5\frac{\partial^2 z}{\partial x\partial y} + 2\frac{\partial^2 z}{\partial y^2} + 7\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} =0 \] is classified as

The general solution of the linear partial differential equation (PDE): \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=z\) is ____ (\(\phi\) is an arbitrary function).

In solving the heat equation using separation of variables, the eigenvalues correspond to ____.

Which of the following statements is/are true for the PDE: \((1+x^2)u_{xx}+(1+y^2)u_{yy}+xu_x+yu_y=0\)?
(I) It is classified as parabolic PDE.
(II) The canonical equation is \(u_{\xi\xi}+u_{\eta\eta}=0\).

Solution of the PDE \(u_x+yu_y=0\) with the initial condition \(u(0,y)=y^3\) is ____.

A string of length 1 meter is fixed at both ends and obeys the wave equation \(u_{tt}=4u_{xx}\) with initial conditions: \(u(x,0)=\sin(\pi x)\), \(u_t(x,0)=0\). Then its solution \(u(x,t)\) is ____.

If \(u = e^{xy}\), then the value of \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\) at (1, 1) is

If \(z = x^2 - y^2\) then \( \frac{1}{x} \frac{\partial z}{\partial x} + \frac{1}{y} \frac{\partial z}{\partial y} = \)

Particular integral of $f(D)y = \cos ax$ is

Complementary function of the differential equation $(D^3 - 8)y = x$ is

Particular integral of $f(D)y = \cos ax$ is

Complementary function of the differential equation $(D^3 - 8)y = x$ is