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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.

Aniruddha Deshmukh
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1 Answers1

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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 $(0,∞)$.

Lutz Lehmann
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  • So, this means that for any $\alpha > 0$, a solution always exists? – Aniruddha Deshmukh Jun 22 '19 at 09:23
  • Yes. But that does not tell if a continuous continuation of the solution to $t=0$ exists, and what values are possible as limit. – Lutz Lehmann Jun 22 '19 at 09:25
  • Are there any references where I can find this? – Aniruddha Deshmukh Jun 22 '19 at 09:27
  • Also, is this true for non linear cases also? Or is it true only for linear equations? – Aniruddha Deshmukh Jun 22 '19 at 09:30
  • Any textbook in the section on linear ODE should contain this either as some theorem or remark. // No, only in the linear cases do you get automatically a global Lipschitz property in $y$ direction over bounded time intervals. In the non-linear case, you get this automatic global existence only for right sides that are sub-linear. – Lutz Lehmann Jun 22 '19 at 09:35
  • I actually found a result here: https://personalpages.manchester.ac.uk/staff/matthias.heil/Lectures/FirstYearODEs/Material/ExistenceAndUniquenessForSecondOrderIVP.pdf – Aniruddha Deshmukh Jun 22 '19 at 09:41