List of practice Questions

Consider the following differential equation: \[ \frac{\partial y}{\partial x} = 3\frac{\partial y}{\partial t} + y \] If \( y(x,0) = 10e^{-2x} \), then the solution of the differential equation is

Identify the following partial differential equations in the given order based on their generic form where \( c \) is a constant, and \( u_1 \) and \( u_2 \) are functions of \( (x, t) \) and \( (x, y) \), respectively:
(i) \[ \frac{\partial^2 u_1}{\partial t^2} = c^2 \frac{\partial^2 u_1}{\partial x^2} \] (ii) \[ \frac{\partial u_1}{\partial t} = c^2 \frac{\partial^2 u_1}{\partial x^2} \] (iii) \[ \frac{\partial^2 u_2}{\partial x^2} + \frac{\partial^2 u_2}{\partial y^2} = 0 \]

A partial differential equation is given below.
\[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0 \]
Possible solution(s) is/are:

The following equation with respect to \(\varphi(x,t)\), where \(a\) is a non-zero constant, represents ________.
\[ \frac{\partial \varphi}{\partial t} + a\frac{\partial \varphi}{\partial x} = 0 \]

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

The general solution of one dimensional wave equation \[ \frac{\partial^2v}{\partial t^2} = c^2 \frac{\partial^2v}{\partial x^2} \] is \(u(x,t)=\)

The complete solution of \[ y^3\frac{\partial u}{\partial x} + x^2\frac{\partial u}{\partial y} = 0 \] is \(u(x,y)=\)

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

The general solution of one dimensional wave equation \[ \frac{\partial^2v}{\partial t^2} = c^2 \frac{\partial^2v}{\partial x^2} \] is \(u(x,t)=\)

The complete solution of \[ y^3\frac{\partial u}{\partial x} + x^2\frac{\partial u}{\partial y} = 0 \] is \(u(x,y)=\)

If \( f \) and \( g \) are the solutions of Laplace equation then \( \nabla \cdot \{(f \nabla g) - (g \nabla f)\ = \)}

If \( f \) and \( g \) are the solutions of Laplace equation then \( \nabla \cdot \{(f \nabla g) - (g \nabla f)\ = \)}

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