Methods | Part IB, 2001

Use the substitution y=xpy=x^{p} to find the general solution of

Lxyd2ydx22x2y=0\mathcal{L}_{x} y \equiv \frac{d^{2} y}{d x^{2}}-\frac{2}{x^{2}} y=0

Find the Green's function G(x,ξ),0<ξ<G(x, \xi), 0<\xi<\infty, which satisfies

LxG(x,ξ)=δ(xξ)\mathcal{L}_{x} G(x, \xi)=\delta(x-\xi)

for x>0x>0, subject to the boundary conditions G(x,ξ)0G(x, \xi) \rightarrow 0 as x0x \rightarrow 0 and as xx \rightarrow \infty, for each fixed ξ\xi.

Hence, find the solution of the equation

Lxy={1,0x<1,0,x>1\mathcal{L}_{x} y= \begin{cases}1, & 0 \leqslant x<1, \\ 0, & x>1\end{cases}

subject to the same boundary conditions.

Verify that both forms of your solution satisfy the appropriate equation and boundary conditions, and match at x=1x=1.

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