Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 1 / viii / Solution

For the initial data, the two weighted integrals equal $({\varphi }_0/c)\mathrm{sgn}(x){\int }_0^L{e}^{\mp fy/c}dy$. Following their opposite characteristics to large time at fixed $x$ gives the additional final-state conditionsThese two scalar constraints determine the two coefficients $A,B$ in the nullspace from part v and therefore make the adjusted solution unique.