Expanding gives $u=2x^2+2c^2t^2$, so $u_{tt}=4c^2$ and $u_{xx}=4$. Hence $u_{tt}=c^2u_{xx}$, proving that it solves the <one-dimensional wave equation>. Equivalently its two summands are oppositely travelling profiles, as in <D'Alembert's formula>.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 6 / Solution

 Solution (source code)