I have a system of ODEs. Can you explain how to solve a system of ODEs using the method Frobenius expansions ?
There are 5 ODEs which are coupled and 5 variables.
$\omega\hat\rho + i\alpha V_z \widehat\rho + i\alpha\rho\widehat {V_z} + \rho\frac{\partial\widehat{V_r}}{\partial r} + in\frac{\rho}{r}\widehat{V_\theta} + \frac{\rho}{r}\widehat{V_r} = 0$
$\omega\widehat{V_r} + i\alpha V_z\widehat{V_r} + \frac{T}{\gamma\rho}\frac{\partial\widehat\rho}{\partial r} + \frac{1}{\gamma}\frac{\partial\widehat T}{\partial r}= 0$
$\omega\widehat{V_\theta} + i\alpha V_z\widehat{V_\theta} + \frac{inT}{\gamma\rho r}\widehat\rho + \frac{in}{\gamma r}\widehat T = 0$
$\omega\widehat{V_z} + \frac{i\alpha T}{\gamma\rho}\widehat\rho + i\alpha V_z\widehat{V_z} + \frac{i\alpha}{\gamma}\widehat T = 0$
$\omega\widehat T + i\alpha(\gamma-1)T\widehat{V_z} + i\alpha V_z\widehat{T} + (\gamma-1)T\frac{\partial\widehat V_r}{\partial r} + (\gamma-1)\frac{T}{r}\widehat{V_r} + in(\gamma-1)\frac{T}{r}\widehat{V_\theta} = 0$
In the above, $\{\widehat{\rho},\widehat{V_r},\widehat{V_\theta},\widehat{V_z},\widehat{T}\}$ are the variables to be solved for in the Frobenius expansion method. $\omega, n, \alpha, \gamma$ are constants.