As we know, for high order linear ODEs with non-constant coefficients we can hardly find an analytical solution, even if we can, the solution might involve very complicated special functions. An alternative way is to seek for numerical solutions.
However, here what I want is a solution $f(x)$ that is "normalized" and non-negative, i.e. \begin{equation} \int_{-\infty}^{\infty} f(x) dx = 1 \end{equation} and $f(x)\geq 0$.
If we try to get a numerical solution using Eular method, we can only specify the boundary conditions instead of forcing the solution to be "normalized" and non-negative.
Here are my questions:
1 How to determine if a non-negative non-zero integrable solution to a given ODE exists?
2 If we know such solution exists, is there any tracktable numerical methods to find a non-negative normalized solution? For example, something based on markov-chain monte-carlo?(I don't know, it's just a guess)
Here is the specific equation that I want to deal with: \begin{equation} a_4 f^{(4)}(x) + a_3 f^{(3)}(x) + a_2 f''(x) + (a_1 + b_1 x^2)f'(x) + (a_0 + b_0 x)f(x) = 0 \end{equation} where $a_0, b_0, a_1, b_1, a_2, a_3, a_4$ are constants.