Paper 2, Section I, A

(a) Find the solution of the differential equation

$y^{\prime \prime}-y^{\prime}-6 y=0$

that is bounded as $x \rightarrow \infty$ and satisfies $y=1$ when $x=0$.

(b) Solve the difference equation

$\left(y_{n+1}-2 y_{n}+y_{n-1}\right)-\frac{h}{2}\left(y_{n+1}-y_{n-1}\right)-6 h^{2} y_{n}=0 .$

Show that if $0<h \ll 1$, the solution that is bounded as $n \rightarrow \infty$ and satisfies $y_{0}=1$ is approximately $(1-2 h)^{n}$.

(c) By setting $x=n h$, explain the relation between parts (a) and (b).

*Typos? Please submit corrections to this page on GitHub.*