I have to solve the following differential equation (that I found in an article):(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\frac{1}{r^{5}}\partial_{r}(r^{5}\partial_{r}h(r)) -E\frac{h(r)}{r^{2}} = - \frac{C}{r^{5}}\delta(r-r_{0})[/itex]

where E and C are two constants.

The authors of the article first find a solution of the previous equation when [itex]r≠r_{0}[/itex] and the result is:

[itex]h(r)=Ar^{c_{\pm}}[/itex] where [itex]c_{\pm} = -2 \pm \sqrt{E+4} [/itex]

Then they have to find the value of the constant A. In order to reach this aim they say that I have to integrate the differential equation and the result should be:

[itex]\frac{C}{2\sqrt{E+4}}\times [/itex] [itex]\frac{1}{r_{0}^{4}}(\frac{r}{r_{0}})^{c_{+}}[/itex] if r ≤ r_{0}

and

[itex]\frac{C}{2\sqrt{E+4}}\times [/itex] [itex]\frac{1}{r^{4}}(\frac{r_{0}}{r})^{c_{+}}[/itex] if r ≥ r_{0}

However I'm not able to recover this result. The point is that when I insert the solution of the homogeneous equation and I integrate the left side of the equation is always equal to zero. Probably I did not understand correctly what is the procedure that I have to follow in order to find the solution to the inhomogeneous equation (once I have the solution of the homogeneous equation), but I do not know how to reach this aim.

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# Find a solution for differential equations

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