Methods | Part IB, 2003

(a) Determine the Green's function G(x,ξ)G(x, \xi) for the operator d2dx2+k2\frac{d^{2}}{d x^{2}}+k^{2} on [0,π][0, \pi] with Dirichlet boundary conditions by solving the boundary value problem

d2Gdx2+k2G=δ(xξ),G(0)=0,G(π)=0\frac{d^{2} G}{d x^{2}}+k^{2} G=\delta(x-\xi), \quad G(0)=0, G(\pi)=0

when kk is not an integer.

(b) Use the method of Green's functions to solve the boundary value problem

d2ydx2+k2y=f(x),y(0)=a,y(π)=b\frac{d^{2} y}{d x^{2}}+k^{2} y=f(x), \quad y(0)=a, y(\pi)=b

when kk is not an integer.

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