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2.15 Wave equation

Verify that

u(t,x)=2\cosh(x+t)-3\sinh(x-t)

satisfies the wave equation

{{\partial^{2}u}\over{\partial x^{2}}}={{\partial^{2}u}\over{\partial t^{2}}}.

Solution. We have $u_{x}=\,{2\sinh(x+t)-3\cosh(x-t)}$ and $u_{t}=\,{2\sinh(x+t)+3\cosh(x-t).}$ Thus

u_{xx}=\,{2\cosh(x+t)-3\sinh(x-t)}\;\;\mbox{and}

u_{tt}=\,{2\cosh(x+t)-3\sinh(x-t).}