Chapter 12.1, Problem 14P

a.

To determine

To Verify- $u\left(x,t\right)=v\left(x+ct\right)+w\left(x-ct\right)$ with any twice differentiable functions $v$ and $w$ satisfies $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$ .

b.

To determine

To Verify- Each $u$ satisfies $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=f\left(x,y\right)$ .

c.

To determine

To Verify- $u=\frac{1}{\sqrt{x^2+y^2+z^2}}$ satisfies $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}=0$ and $u=\ln \left(x^2+y^2\right)$ satisfies $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$ .

d.

To determine

To Verify- $u$ with any $v$ and $w$ satisfies the given PDE.