Verify for each of the following differential equations that given function or functions are solutions:

$\textbf{(a)}$

\begin{align*}\displaystyle

u_{xx}+u_{yy}&=0\\

u_1(x,y)&=x^2-y^2\\

u_2(x,y)&=\cos{x}cosh{y}

\end{align*}

$\textbf{(b) $\lambda$ a real constant }$

\begin{align*}\displaystyle

a^2u_{xx}&=u_t\\

u_1(x,t)&=e^{-\alpha^2t}\sin{x}\\

u_2(x,t)&=e^{-\alpha^2\lambda^2t}\sin{\lambda x}

\end{align*}

$\textbf{(c) $\lambda$ a real constant }$

\begin{align*}\displaystyle

a^2u_{xx}&=u_{tt}\\

u_1(x,t)&=\sin{\lambda x}\sin{\lambda \alpha t}\\

u_2(x,t)&=\sin{(x-\alpha t)}

\end{align*}

ok I have never done this before and just trying to learn as much as I can before I take the class

I read the chapter but it was kinda ???

much thank you ahead..