I tried to solve this differential equation:
$$y''+\frac{1}{x}y'-\frac{9}{x^2}y=0\text{, where }x \in (1,2), y'(1)=\alpha, y(2)=\beta$$
I should use shooting method but I have a problem with boundaries because it is a mix of Neumann and Dirichlet boundary condition. Does anyone know how to transfer it only to Dirichlet boundary condition or how to solve it?
Thank you.
With $y = x^r$ we get that
$$r(r-1)x^{r-2} + rx^{r-2} - 9x^{r-2} = 0 \implies r^2-9 = 0$$
which means our two linearly independent solutions are
$$y = Ax^3 + Bx^{-3}$$
Plugging in the initial conditions gives us
$$\begin{cases}\alpha = 3A - 3B \\ \beta = 8A + \frac{1}{8}B\\ \end{cases} \implies \begin{cases}A = \frac{24\beta+\alpha}{195} \\ B = \frac{24\beta-64\alpha}{195} \\ \end{cases}$$