First order linear system of differential equations

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.

• 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 $A$? – BAYMAX Sep 28 '17 at 4:34
• Also this reference might help! – BAYMAX Sep 28 '17 at 4:42
• Thank You this method not works I think the problem is more difficult – Fethi Sep 28 '17 at 6:49
• Yes,I too tried here – BAYMAX Sep 28 '17 at 9:27
• 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. – MrYouMath Sep 29 '17 at 10:23

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