Willow Pratt

2022-07-10

For some research Im doing, I've derived an equation of the form below for $C\left(r\right)$
${C}^{″}+\frac{2}{r}{C}^{\prime }=W+\frac{f}{C}$
Or, if you prefer,
$C{C}^{″}+\frac{2}{r}C{C}^{\prime }-W\cdot C=f$
This has the form of a 2nd order inhomogeneous linear equation with variable coefficients but the problem for me here is I don't know a clever way to solve this as C appears throughout equation. Any ideas on how to solve this and indeed, if there is even a solution ?
Id plug it into mathematica but am out of office travelling so would be most grateful for input!

Jordan Mcpherson

Expert

Hint:
Let $U=rC$,
Then $\frac{dU}{dr}=r\frac{dC}{dr}+C$
$\frac{{d}^{2}U}{d{r}^{2}}=r\frac{{d}^{2}C}{d{r}^{2}}+\frac{dC}{dr}+\frac{dC}{dr}=r\frac{{d}^{2}C}{d{r}^{2}}+2\frac{dC}{dr}$
$\therefore \frac{1}{r}\frac{{d}^{2}U}{d{r}^{2}}=W+\frac{fr}{U}$
$\frac{{d}^{2}U}{d{r}^{2}}=Wr+\frac{f{r}^{2}}{U}$
You can consider as two members Emden-Fowler type nonlinear ODE

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