Paper 2, Section I, A

Differential Equations | Part IA, 2016

(a) Find the solution of the differential equation

yy6y=0y^{\prime \prime}-y^{\prime}-6 y=0

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

(b) Solve the difference equation

(yn+12yn+yn1)h2(yn+1yn1)6h2yn=0.\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<h10<h \ll 1, the solution that is bounded as nn \rightarrow \infty and satisfies y0=1y_{0}=1 is approximately (12h)n(1-2 h)^{n}.

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

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