2.II.6D2 . \mathrm{II} . 6 \mathrm{D} \quad

Differential Equations | Part IA, 2003

Define the Wronskian W(x)W(x) associated with solutions of the equation

d2ydx2+p(x)dydx+q(x)y=0\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0

and show that

W(x)exp(xp(ξ)dξ).W(x) \propto \exp \left(-\int^{x} p(\xi) d \xi\right) .

Evaluate the expression on the right when p(x)=2/xp(x)=-2 / x.

Given that p(x)=2/xp(x)=-2 / x and that q(x)=1q(x)=-1, show that solutions in the form of power series,

y=xλn=0anxn(a00)y=x^{\lambda} \sum_{n=0}^{\infty} a_{n} x^{n} \quad\left(a_{0} \neq 0\right)

can be found if and only if λ=0\lambda=0 or 3 . By constructing and solving the appropriate recurrence relations, find the coefficients ana_{n} for each power series.

You may assume that the equation is satisfied by y=coshxxsinhxy=\cosh x-x \sinh x and by y=sinhxxcoshxy=\sinh x-x \cosh x. Verify that these two solutions agree with the two power series found previously, and that they give the W(x)W(x) found previously, up to multiplicative constants.

[Hint: coshx=1+x22!+x44!+,sinhx=x+x33!+x55!+.]\left.\cosh x=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\ldots, \quad \sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\ldots .\right]

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