# How can I solve for these partial derivatives given a set of variables

#### Boltzman Oscillation

Problem Statement
given the one dimensional wave equation and the two variables E and n, solve for:
Relevant Equations
$$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$$
E = x + ct
n = x - ct
I am given the following:
$$u = (x,t)$$
$$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$$
and
$$E = x + ct$$
$$n = x - ct$$
I need to solve for $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}{\partial t^2}$$
using the chain rule.How would I even begin? Would I have to say that:
$$x = (E, t)$$
$$x =(n, t)$$
thus
$$u(x,t) = u( x(E,t) , t)$$
and
$$u(x,t) = u(x(n,t),t)$$?

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#### PeroK

Science Advisor
Homework Helper
Gold Member
2018 Award
Your notation is struggling here. Normally the idea with the wave equation is to let $f$ be any scalar function. And define:

$u(x, t) = f(E) = f(x + ct)$

Now, does this $u(x, t)$ satisfy the wave equation?

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