# An Ordinary Differential Equation with time varying coefficients

Let $A$ and $B$ be complex numbers, let $\beta_1$ be real and $\beta_2=2$. Consider a following Ordinary Differential Equation: $$\frac{ d^2 r_t}{ d t^2} + \left(\frac{A}{t^{\beta_1}} + \frac{B}{t^{\beta_2}}\right) r_t = 0$$ By substituting for $t^{2 - \frac{\beta_1+\beta_2}{2}}$ and using the power series method I have shown that the fundamental solutions to this equation are: $$r_t = \sqrt{t} J_{\pm \frac{\sqrt{1-4 B}}{-2 + \beta_1}} \left[\frac{2 \sqrt{A} t^{1-\frac{\beta_1}{2}}}{-2+\beta_1}\right]$$ Here $J_\beta$ is the Bessel function. Unfortunately the power series method fails in case of $\beta_2 \neq 2$ . Is there a different method for finding the solution in case $\beta_2 \neq 2$.