Methods | Part IB, 2004

Obtain the Green function G(x,ξ)G(x, \xi) satisfying

G+2xG+k2G=δ(xξ),G^{\prime \prime}+\frac{2}{x} G^{\prime}+k^{2} G=\delta(x-\xi),

where kk is real, subject to the boundary conditions

G is finite  at x=0,G=0 at x=1.\begin{array}{rll} G \text { is finite } & \text { at } & x=0, \\ G=0 & \text { at } & x=1 . \end{array}

[Hint: You may find the substitution G=H/xG=H / x helpful.]

Use the Green function to determine that the solution of the differential equation

y+2xy+k2y=1,y^{\prime \prime}+\frac{2}{x} y^{\prime}+k^{2} y=1,

subject to the boundary conditions

y is finite  at x=0,y=0 at x=1,\begin{array}{rll} y \text { is finite } & \text { at } & x=0, \\ y=0 & \text { at } & x=1, \end{array}


y=1k2[1sinkxxsink]y=\frac{1}{k^{2}}\left[1-\frac{\sin k x}{x \sin k}\right]

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