If we have a second order ODE
$$y'' + \dfrac{\alpha}{t} y' – \dfrac{1}{t^{\alpha}} y = 0$$
where $\alpha > 0$ is a real number, can we have some results on the existence of the solutions for $t \in \left( 0,\infty \right)$? Or can we directly find the solutions?
I know that if $\alpha \leq 2$, then $t = 0$ is a regular singular point and otherwise it is irregular.
For regular singular points, I know we can use the Frobebius method to find the solutions. For the same, I also asked a question about finding the power series of $\dfrac{1}{t^{\alpha}}$ here: Power series for an arbitrary power of a variable.
Best Answer
All solutions starting on the positive half axis extend to $(0,∞)$.
This follows from the general result that the maximal domain of solutions of explicit linear ODE is (at least) the maximal open interval containing the initial point on which the coefficient functions are continuous. $\fracαt$ and $-\frac1{t^α}$ are continuous on $(0,∞)$.