Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 14A (Methods)

Substitution of $y=e^{-x}$ givesFor a second solution $y=ax+b$, the ordinary differential equation reduces to $a\lambda -b=0$. Taking $a=1$ gives the linearly independent solution

The solution $x+\lambda$ satisfies the left boundary condition, while $e^{-x}$ satisfies decay at infinity. The proposed expression for $x<\xi$ is therefore the correct left homogeneous solution. For $x>\xi$, write $G=C{e}^{-x}$. Continuity at $x=\xi$ givesHence the Green function isIndeed, its derivative has the required unit jump,which produces the Dirac delta distribution in $L[G]$.

For $\lambda =2$, the Green representation of the boundary value problem isSplitting the integral at $\xi =x$ givesThereforeIt satisfies $y(0)=2y^{\prime }(0)$ and tends to zero as $x\to \infty$.