Patatiniuh

2022-07-12

I want to solve the following first order linear system of differential equations:
$\begin{array}{rl}& \frac{d{y}_{0}}{dx}+x{y}_{1}=\lambda {y}_{2}\\ & \frac{d{y}_{1}}{dx}+x{y}_{2}=\lambda {y}_{0}\\ & \frac{d{y}_{2}}{dx}+\frac{\alpha }{x}{y}_{2}+x{y}_{0}=\lambda {y}_{1}\end{array}$
with intial conditions: ${y}_{0}\left(0\right)=1,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{1}\left(0\right)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{2}\left(0\right)=0.$

Nathen Austin

After putting your system into the form $\stackrel{˙}{\mathbit{y}}\left(x\right)=\mathbit{A}\left(x\right)\mathbit{y}\left(x\right)$ you will see that the system is a linear time variant (LTV) differential equation. The system matrix is given by
$\mathbit{A}\left(x\right)=\left[\begin{array}{ccc}0& -x& \lambda \\ \lambda & 0& -x\\ -x& \lambda & -\alpha /x\end{array}\right]$
your state vector is $\mathbit{y}=\left[{y}_{0},{y}_{1},{y}_{2}{\right]}^{T}$.
Solving this system is not trivial. I gave this ODE to Maple but it was not able to solve it. You could also try Mathematica/Sympy/Mupad or other computer algebra systems.
There is a symbolic solution to LTV systems referred as the Peano-Baker series. In some cases, you might be able to find a closed form solution from this series, but it is not very likely for your problem.
If you just want to study the stability of some solutions then you might also consider using Floquet-Theory.

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