Differential Equations | Part IA, 2004

Two solutions of the recurrence relation

xn+2+b(n)xn+1+c(n)xn=0x_{n+2}+b(n) x_{n+1}+c(n) x_{n}=0

are given as pnp_{n} and qnq_{n}, and their Wronskian is defined to be

Wn=pnqn+1pn+1qnW_{n}=p_{n} q_{n+1}-p_{n+1} q_{n}

Show that

Wn+1=W1m=1nc(m)W_{n+1}=W_{1} \prod_{m=1}^{n} c(m)

Suppose that b(n)=αb(n)=\alpha, where α\alpha is a real constant lying in the range [2,2][-2,2], and that c(n)=1c(n)=1. Show that two solutions are xn=einθx_{n}=e^{i n \theta} and xn=einθx_{n}=e^{-i n \theta}, where cosθ=α/2\cos \theta=-\alpha / 2. Evaluate the Wronskian of these two solutions and verify ()(*).

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