# Green's function for Cauchy-Euler equidimensional equation

1. May 20, 2010

### anubhab

Hi,

I am trying to compute the Green's function for a Cauchy-Euler equidimensional equation,
$$\frac{d^2G}{dx^2}+\frac{a}{(x-x_c)^2}G=A_1\delta(x-x')$$
If the impulse is located at a location $$x'\neq x_c$$ then computation of Green's function is not an issue. What happens when $$x'= x_c$$ ?
$$\frac{d^2G}{dx^2}+\frac{a}{(x-x_c)^2}G=A_1\delta(x-x_c)$$
The solution of the homogeneous equation is $$(x-x_c)^{1/2\pm\nu}$$ where, $$\nu=\sqrt{1/4-a}$$
The trouble is if one tries to relate the change in slope of the Green's function with the strength of the impulse one has,
$$\int_{x_c-\epsilon}^{x_c+\epsilon}\frac{d^2G}{dx^2} dx+\int_{x_c-\epsilon}^{x_c+\epsilon}\frac{a}{(x-x_c)^2}G \!dx=A_1$$
From the Frobenius exponents we see that the only for $$a=0$$ is the second integral is a Cauchy principal value integral.
Is it sensible to seek for a Green's function for such cases or one needs to make certain modification?
Thank you,

anubhab