Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 5 / a / ii

The characteristic curves satisfyIntegrating suggests the characteristic coordinatesTheir Jacobian determinant is $2/x$, so this is an invertible local change of variables whenever $x\ne 0$. Write $u(x,y)=U(\xi ,\eta )$. The chain rule givesSubstitution cancels both unmixed second derivatives and both first derivatives:The canonical form of a second-order partial differential equation is thereforeThe cancellation of the first derivatives is the key feature of these logarithmic characteristic coordinates.