Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 10

With $x=e^t$, the chain rule gives $x{y}^{\prime }=\dot{y}$ and $x^2{y}^{\prime \prime }=\ddot{y}-\dot{y}$. The Euler differential equation becomesIts characteristic polynomial has roots $-1\pm 2\sqrt{2}$, so, in addition to the requested transformation, its general solution is $y(x)=C_1{x}^{-1+2\sqrt{2}}+C_2{x}^{-1-2\sqrt{2}}$ on $x>0$.

Assume $v\ne 0$ so that the change of variables is invertible. The chain rule gives ${\partial }_x={\partial }_r+{\partial }_s$ and ${\partial }_t=v({\partial }_r-{\partial }_s)$. Commuting the mixed partial derivatives for a twice continuously differentiable solution givesThus the wave equation is $F_{rs}=0$. Integrating once says that $F_r$ depends only on $r$; integrating again givesThe two arbitrary twice differentiable functions are waves moving in opposite directions at speed $|v|$. This is D'Alembert's formula, on a product domain in $(r,s)$ or globally on the usual full line.

Integrating the mixed partial derivative successively gives $z=x^3{y}^2/6+A(x)+B(y)$. The first boundary value gives $A(x)=x^2-1-B(0)$. Substitution at $x=1$ then gives $B(y)-B(0)=\sin y-y^2/6$. ThereforeIts mixed derivative is $x^{2}y$, and direct evaluation checks both boundary values. Their common corner is compatible: both prescribe $z(1,0)=0$.