Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 2 / b / Solution

Write $z(x,y)=Z(r,s)$ in the new coordinates. Sincethe partial differential equation becomesFor a twice continuously differentiable solution of a differential equation, $Z_r$ is independent of $s$, so integration first in $r$ and then in $s$ givesThis is the decomposition of the one-dimensional wave equation into waves traveling in opposite directions.

The first initial condition gives $\varphi (y)+\psi (y)=\cos y$. The second initial condition gives ${\varphi }^{\prime }(y)-{\psi }^{\prime }(y)=0$. Differentiating the first relation and combining the two relations yields ${\varphi }^{\prime }=ps{i}^{\prime }=-\frac{1}{2}\sin y$. Thus $\varphi (y)=\frac{1}{2}\cos y+C$ and $\psi (y)=\frac{1}{2}\cos y-C$. The constants cancel, and the cosine addition formula givesBoth initial conditions and the partial differential equation follow directly by differentiation.