Verifying that a function solves a PDE

Prove that the function

($a$ and $b$ are constants) satisfies the Laplace equation

Prove that the function

($a$ and $b$ are constants) satisfies the heat conduction equation

Prove that the function

(where $r=\sqrt{(x-a)^{2}+(y-b)^{2}+(z-c)^{2}}$), when $r \neq 0$, satisfies the Laplace equation

Prove that the function

(where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and $C_{1}, C_{2}$ are constants) satisfies the Helmholtz equation

Prove that the function

(where $f$ is an arbitrary differentiable function) satisfies the equation

Prove that

(where $f$ is an arbitrary differentiable function) satisfies the equation

Verify the following identity: $y\dfrac {\partial z}{\partial x}-x\dfrac {\partial z}{\partial y}=0$, if $z=\varphi (x^{2}+y^{2})$.

Verify the following identity: $x^{2}\dfrac {\partial z}{\partial x}-xy\dfrac {\partial z}{\partial y}+y^{2}=0$, if $z=\dfrac {y^{2}}{3x}+\varphi (xy)$.

Verify the following identity: $(x^{2}-y^{2})\dfrac {\partial z}{\partial x}+xy\dfrac {\partial z}{\partial y}=xyz$, if $z=e^{x}\varphi \left(ye^{\frac{x^{2}}{2y^{2}}}\right)$.

Verify the following identity: $x\dfrac {\partial u}{\partial x}+\alpha y\dfrac {\partial u}{\partial y}+\beta z\dfrac {\partial u}{\partial z}=nu$, if $u=x^{n}\varphi \left(\dfrac {y}{x^{\alpha }},\dfrac {z}{x^{\beta }}\right)$.

Verify the following identity: $x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=u+\dfrac {xy}{z}$, if

Verify the following identity: $\dfrac {\partial^{2} u}{\partial t^{2}}=a^{2}\dfrac {\partial^{2} u}{\partial x^{2}}$, if $u=\varphi (x-at)+\psi (x+at)$.

Verify the following identity: $\dfrac {\partial^{2} u}{\partial x^{2}}-2\dfrac {\partial^{2} u}{\partial x \partial y}+\dfrac {\partial^{2} u}{\partial y^{2}}=0$, if

Verify the following identity: $x^{2}\dfrac {\partial^{2} u}{\partial x^{2}}+2xy\dfrac {\partial^{2} u}{\partial x \partial y}+y^{2}\dfrac {\partial^{2} u}{\partial y^{2}}=0$, if

Verify the following identity: $x^{2}\dfrac {\partial^{2} u}{\partial x^{2}}+2xy\dfrac {\partial^{2} u}{\partial x \partial y}+y^{2}\dfrac {\partial^{2} u}{\partial y^{2}}=n(n-1)u$, if

Verify the following identity: $\dfrac {\partial u}{\partial x}\dfrac {\partial^{2} u}{\partial x \partial y}=\dfrac {\partial u}{\partial y}\dfrac {\partial^{2} u}{\partial x^{2}}$, if $u=\varphi \left[x+\psi (y)\right]$.

Prove that the function $z=z(x,y)$ defined by the equation

[where $\Phi (u,v)$ is an arbitrary differentiable function of the variables $u$, $v$, and $a$ and $b$ are constants] is a solution of the equation

Describe the geometric nature of the surface (1).

Prove that the function $z=z(x,y)$ defined by the equation

[where $\Phi (u,v)$ is an arbitrary differentiable function of the variables $u$ and $v$] satisfies the equation

Describe the geometric nature of the surface (2).

Prove that the function $z=z(x,y)$ defined by the equation

[where $\Phi (u)$ is an arbitrary differentiable function of the variable $u$, and $a$, $b$ and $c$ are constants] satisfies the equation

Describe the geometric nature of the surface (3).

The function $z=z(x,y)$ is given by the equation

Prove that:

The function $z=z(x,y)$ is given by the equation

Prove that:

Prove that the function $z=z(x,y)$ defined by the system of equations

[where $\alpha =\alpha (x,y)$ is a parameter and $f(\alpha )$ is an arbitrary differentiable function] satisfies the equation

Prove that the function $z=z(x,y)$ given by the system of equations

satisfies the equation

Prove that the function $z=z(x,y)$ defined by the system of equations

satisfies the equation

Prove that the function $z=z(x,y)$ given by the system of equations

satisfies the equation

Prove that the implicit function $z=z(x,y)$ defined by the equation $y=x\varphi (z)+\psi (z)$ satisfies the equation

Let $f(x)$ be a twice-differentiable function and $F(x)$ a differentiable function. Prove that the function

satisfies the wave equation for a vibrating string

together with the initial conditions: $u(x,0)=f(x)$, $u'_{t}(x,0)=F(x)$.

Let $f(x,y)$ be continuous; prove that the function

satisfies the equation

Verify that $\frac{\partial u}{\partial t} = a^{2}\frac{\partial^{2} u}{\partial x^{2}}$, if $u = \frac{1}{\sqrt{t}} e^{-x^{2}/(4a^{2}t)}$.

Verify that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z} = 0$, if $u = \left(\frac{x}{y}\right)^{z/y}$.

Compute $\frac{\partial^{2} u}{\partial x^{2}}-2\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}$, if $u = \varphi (x+y)$.

Compute $x^{2}\frac{\partial^{2} u}{\partial x^{2}}-2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}}$, if $u = f\left(\sqrt{x^{2}+y^{2}}\right)$.

Compute $\frac{\partial^{2} u}{\partial x^{2}} \cdot \frac{\partial^{2} u}{\partial y^{2}}-\left(\frac{\partial^{2} u}{\partial x \partial y}\right)^{2}$, if $u = \varphi (xy)$.

The function called the potential of the ball $x^{2}+y^{2}+z^{2} = a^{2}$ is defined by the equalities:

Verify that $u$ and its first derivatives are continuous for any $x$, $y$, and $z$, and that the Laplacian

depending on whether the point $M(x,y,z)$ lies outside the ball $x^{2}+y^{2}+z^{2} = a^{2}$ or inside it.

Prove that if $f(u)$ is an arbitrary differentiable function, then the function $\varphi (x ; y)$ satisfies the given equation:

Prove that if $f(u ; v)$ is an arbitrary differentiable function, then the function $\varphi (x ; y ; z)$ satisfies the given equation:

Prove that:

Prove that the function

satisfies the $n$-dimensional Laplace equation

Prove that the function

where $r = \sqrt{x^{2}+y^{2}+z^{2}}, k, C_{1}, C_{2}$ are constants, satisfies the Helmholtz equation

Prove that:

Prove that the function

satisfies the $n$-dimensional heat equation

Prove that the function $u(t ; x) = \frac{1}{\sqrt{t}} e^{i x^{2} /(4 t)}$ satisfies the Schrödinger equation

Prove that the one-dimensional wave equation

is satisfied by the following functions:

Find a function $u(t ; x)$ satisfying the conditions

Prove that the function $u(t ; x ; y) = \frac{1}{\sqrt{a^{2} t^{2}-x^{2}-y^{2}}}$ satisfies the two-dimensional wave equation

Let $f$ and $g$ be arbitrary twice differentiable functions. Prove that the function $u$ satisfies the given equation:

Prove that if the equation $y = x f(u)+g(u)$, where $f$ and $g$ are twice differentiable functions, defines a twice differentiable function $u(x ; y)$, then it satisfies the equation

Let the function $f(t ; x)$ be continuous on the plane. Prove that the function

satisfies the wave equation $\frac{\partial^{2} u}{\partial t^{2}}-a^{2} \frac{\partial^{2} u}{\partial x^{2}} = f(t ; x)$.