Write the d’Alembert solution for the problem $u_{t t}=c^{2} u_{x x}$ for $-\infty 0$ and $u(x, 0)=f(x), u_{t}(x, 0)=g(x)$ for $-\infty<x<\infty$. $c=1, f(x)=x^{2}, g(x)=-x$

Write the d’Alembert solution for the problem $u_{t t}=c^{2} u_{x x}$ for $-\infty<x<\infty, t>0$ and $u(x, 0)=f(x), u_{t}(x, 0)=g(x)$ for $-\infty<x<\infty$ . $c=1, f(x)=x^{2}, g(x)=-x$

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We are solving the equation (wave equation with $c^2=1$ )

\dfrac{\partial^2 u}{\partial t^2}= \dfrac{\partial^2 u}{\partial x^2}

usin the d'Alambert method. We employ the substitution

$\xi=x-t$ \ and \ $\eta=x+t$

expressing $x$ and $t$ in terms of $\xi$ and $\eta$ that is

$x=\dfrac{1}{2}(\eta+\xi)$ \ and \ $t=\dfrac{1}{2}(\eta-\xi)$

We use this to express the derivative in terms of $\xi$ and $\eta$ to get (after some arduous computation)

\dfrac{\partial^2 u}{\partial x^2}=\dfrac{\partial^2 u}{\partial \xi^2}+2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}

and

\dfrac{\partial^2 u}{\partial t^2}= \dfrac{\partial^2 u}{\partial \xi^2}-2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}

Insertin this into our initial equation we get

\dfrac{\partial^2 u}{\partial \xi^2}-2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}=\dfrac{\partial^2 u}{\partial \xi^2}+2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}

we cancel (what's the same) from the left and right

\dfrac{\partial^2 u}{\partial \xi \partial \eta}=0

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