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Suppose I have a differential equation

\begin{align} \sum_{i=0}^4 f_i (x) y^{(i)}(x) = 0, \end{align} where $y^{(i)}(x)$ represents the $i$th derivative of $y(x)$ and the coefficient functions $f_i(x)$ are all polynomials of the variable $x$. Suppose also there are some boundary conditions, although for the purposes of this question they may not be important.

For special cases of the $f_i$'s, there are exact analytic solutions to this differential equation. However, in general, there are no exact solutions that can be found.

Are there any techniques to arriving at an approximate solution $y(x)$ with arbitrary polynomial coefficient functions $f_i(x)$?

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  • $\begingroup$ Have you considered the power series method ? $\endgroup$ – Yves Daoust Jan 23 at 23:09
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I suggest to have a look to the maple command Slode . This command (package) is dedicated to solving linear ODE with polynomial coefficient. I just solved a fourth order ODE with polynomial coefficient of second order with this package and it seems ok. Hope that this answer will be helpful

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