Methods | Part IB, 2005

Consider the differential equation for x(t)x(t) in t>0t>0

x¨k2x=f(t),\ddot{x}-k^{2} x=f(t),

subject to boundary conditions x(0)=0x(0)=0, and x˙(0)=0\dot{x}(0)=0. Find the Green function G(t,t)G\left(t, t^{\prime}\right) such that the solution for x(t)x(t) is given by

x(t)=0tG(t,t)f(t)dt.x(t)=\int_{0}^{t} G\left(t, t^{\prime}\right) f\left(t^{\prime}\right) d t^{\prime} .

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