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: ${y}_{0}(0)=1,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{1}(0)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{2}(0)=0.$

$\begin{array}{rl}& \frac{d{y}_{0}}{dx}+x{y}_{1}=\lambda {y}_{2}\\ & \frac{d{y}_{1}}{dx}+x{y}_{2}=\lambda {y}_{0}\\ & \frac{d{y}_{2}}{dx}+\frac{\alpha}{x}{y}_{2}+x{y}_{0}=\lambda {y}_{1}\end{array}$

with intial conditions: ${y}_{0}(0)=1,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{1}(0)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{2}(0)=0.$