how could I solve the linear differential equation $$ -y''(x)+x^{3}y(x)=0$$ with the boundary conditions $ y(0)=y(\infty) $ by the linear shooting method ?
If we had $ y(0)=y(1)=0$ then the interval is finite however how can i deal with $ y(\infty) =0 $ ? i guess that we can get a large number $ L \rightarrow 10^{6} $ and solve it $ y(0)=y(L)=0$ however I do not know what is the best method to solve numerically my eigenvalue problem.
First, it is not necessarily sufficient to choose a large number because it is large. Rather, you will want to choose a number $L$ such that $y''(L) \approx y'(L) \approx 0$. In some cases, it is sufficient to replace $\infty$ with something as small as $5$ (such as computing the boundary layer velocity profile over a flat plate). Other cases, you may need a number quite large. To estimate a sufficient value, make a guess for $y'(0)$, and integrate. Find an acceptable $L$ such that you achieve the desired error tolerance.
Second, assume you can find such an $L$. There are a number of ways to implement a shooting method. One way is to re-frame the problem as a "root finding" algorithm. First, you guess a value $c$ and let $y'(0)=c$. Then, integrate the ODE using whatever method suits your needs. Then, you compute the error $||y(L)-0|| = e$ and compute a new value of $c$. You can do this using Newton's method, for example, or via bisection.
One option would be to render the point $x=\infty$ finite by some coordinate transformation. This transformation will necessarily be singular, so you need to make sure you treat the singular point appropriately.
