A general form of second order differential equations is written as
for \(a<x<b\) with boundary conditions
The solution to it is
where \(G(x\vert \xi)\) is the Green’s function. The Green’s function is an impulse response of the dynamical system, i.e.,
with the two boundary conditions,
First order differential of Green’s function \(G'(x\vert \xi)\) has a jump discontinuity at \(x=\xi\). This is expected since we have to have a dirac delta type of response at \(x=\xi\).
For a 2nd order differential equation,
the Green function is
We already know the two solutions to the homogeneous equation, \(y=1\) or \(y=x\). However, only the second solution can satisfy the boundary conditions. Then the Green’s function should have these properties,
The boundary conditions give us
We also know that the Green’s functon must be continuous, we then require
Using the discontinuity of the first order derivative of the Green’s function, we get
With all these equation, we determine the Green’s function,
To get the solution to \(y\), we integrate over \(\xi\),
It is that easy. This is the super power of the Green’s function.