Question

Show that: 1) u = e^x sin y is a solution of Laplace's equation.
2) u = x^2 + t^2 is a solution of the wave equation.

Answer 1

50744, Physics, Optics

Question Show that: 1) $u=e^{x}\sin y$ is a solution of Laplace’s equation. 2) $u=x^{2}+t^{2}$ is a solution of the wave equation.

Solution 1.Laplace equation is

$\Delta f=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}=0$

Let us substitute given function. We get

$\frac{\partial^{2}(e^{x}\sin y)}{\partial x^{2}}+\frac{\partial^{2}(e^{x}\sin y)}{\partial y^{2}}=e^{x}\sin y+\frac{\partial(e^{x}\cos y)}{\partial y}=e^{x}\sin y-e^{x}\sin y=0$

So, it satisfies Laplace equation.

2. Wave equation is

$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$

Let us substitute given function. We get

$\frac{\partial^{2}(x^{2}+t^{2})}{\partial t^{2}}-c^{2}\frac{\partial^{2}(x^{2}+t^{2})}{\partial x^{2}}=$

$=2\frac{\partial t}{\partial t}-c^{2}\frac{\partial x}{\partial x}=1-c^{2}=0$

So, it satisfies wave equation, if $c^{2}=1$.