Recently, a friend challenged me to find the general solution of the following differential equation: $$\frac{d^2 y}{dx^2}+(x+1)\cdot \frac{dy}{dx}+5x^2\cdot y=0 \tag{1}$$ This is a second-order linear ordinary differential equation. I have tried putting this ODE into the form of a Sturm-Liouville Equation by multiplying both sides by $e^{\int (x+1)~dx}$ to obtain: $$e^{\frac{x^2}{2}+x}\cdot\frac{d^2 y}{dx^2}+(x+1)\cdot e^{\frac{x^2}{2}+x}\cdot \frac{dy}{dx}+5x^2\cdot e^{\frac{x^2}{2}+x}\cdot […]

This question is related to another one I asked earlier here. For reference, I asked for help writing a generalized Fourier Series for the function $f(x) = 1$ for $0<x<1$, in terms of the eigenfunctions $\displaystyle w_{n} = \cos \left[\left( n – \frac{1}{2}\right)\pi x \right]$, $n = 0,1,2, \cdots$ for the Sturm-Liouville problem: $\begin{matrix} – […]

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