Having an issue differentiating the following equation for purposes of reducing into two 1st order ODE's. Where: $y(1)=0$, $y'(1) = 2/3$ and $h = 0.5$
$$3x^2y'' - 5xy'+5y = 0$$
when making $y'=z$.
Trying to find $dy/dx$ and $dz/dx$, then put it into a 4th order Runge-Kutta where:
$$dy/dx=f(x,y,z)$$ $$dz/dx=g(x,y,z)$$
This is my process (roughly): Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.
If $z = y'$ then $z' = y''$, and you get \begin{cases} 3x^2z' - 5xz+5y = 0\\ \phantom{3x^2z' - 5xz+5}y' = z. \end{cases}
We can rewrite this by solving for $z'$ and $y'$ as \begin{cases} z' = \frac{5xz - 5y}{3x^{2}}\\ y' = z. \end{cases}