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I want to solve the following first order linear system of differential equations :

\begin{align} &\frac{dy_0}{dx}+xy_1=\lambda y_2\\& \frac{dy_1}{dx}+xy_2=\lambda y_0\\& \frac{dy_2}{dx}+\frac{\alpha}{x}y_2+xy_0=\lambda y_1 \end{align} with intial conditions: $y_0(0)=1, \,\, y_1(0)=0,\,\,y_2(0)=0.$

Thank you.

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  • $\begingroup$ we can write the above in the form $\dot{Y} =A Y$ , the solutions to this is of the form $Y(x) = \eta e^{rx}$ where $\eta , r$ are the eigenvectors and eigenvalues of matrix $A$? $\endgroup$ – BAYMAX Sep 28 '17 at 4:34
  • $\begingroup$ Also this reference might help! $\endgroup$ – BAYMAX Sep 28 '17 at 4:42
  • $\begingroup$ Thank You this method not works I think the problem is more difficult $\endgroup$ – Fethi Sep 28 '17 at 6:49
  • $\begingroup$ Yes,I too tried here $\endgroup$ – BAYMAX Sep 28 '17 at 9:27
  • $\begingroup$ I removed the $(x)$ dependencies in your differential equations so that it is easier to read. Feel free to change it if you think that was important. $\endgroup$ – MrYouMath Sep 29 '17 at 10:23
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After putting your system into the form $\dot{\boldsymbol{y}}(x)=\boldsymbol{A}(x)\boldsymbol{y}(x)$ you will see that the system is a linear time variant (LTV) differential equation. The system matrix is given by

$$\boldsymbol{A}(x)=\begin{bmatrix}0 & -x & \lambda\\ \lambda & 0 & -x\\-x & \lambda & -\alpha/x \end{bmatrix}$$

your state vector is $\boldsymbol{y}=[y_0,y_1,y_2]^T$.

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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