Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 3 / 7A (Methods)

The substitution removes the zero-order term: $u_x=e^{-x^2}(v_x-2xv)$ and $u_y=e^{-x^2}{v}_y$, so the partial differential equation becomes $v_x+x{v}_y=1$. Its method of characteristics givesThe invariant is $\eta =y-x^2/2$. Therefore $v=x+F(\eta )$. The initial line $x=0$ is noncharacteristic, and the boundary data give $F(\eta )=\eta e^{-{\eta }^2}$. HenceEach characteristic meets the data line once, ensuring uniqueness among differentiable solutions, and substituting the result verifies both the equation and the data.