Solution to a higher order ordinary differential equation

Question

Let $q$ be a strictly positive integer and let $\beta \neq q$ be a real number. Consider a following Ordinary Differential Equation(ODE): \begin{equation} \frac{d^q r_t}{d t^q} + \frac{1}{t^\beta} r_t = 0 \end{equation} Using Mathematica I have checked that the fundamental solutions to that ODE reads: \begin{equation} r_t^{(\eta)} = t^\eta F_{0,q-1}\left[\begin{array}{c} \{\} \\ \left\{ 1+ \frac{\xi}{q-\beta} \right\}_{\xi=1}^\eta \cup \left\{ 1- \frac{\xi}{q-\beta} \right\}_{\xi=1}^{q-1-\eta} \end{array}; -\frac{t^{q-\beta}}{(q-\beta)^q}\right] \end{equation} where $\eta=0,\cdots,q-1$. Here $F_{p,q}$ is the hypergeometric function. The quantity on the bottom of the square brackets is understood as a concatenation of two sequences the first one of length $\eta$ and the second one of length $q-1-\eta$. In the special case $q=2$ the fundamental solutions read: \begin{equation} \left( \sqrt{t} J_{\frac{1}{-2+\beta}}\left[\frac{t^{1-\frac{\beta}{2}}}{\frac{\beta}{2}-1}\right] , \sqrt{t} J_{\frac{-1}{-2+\beta}}\left[\frac{t^{1-\frac{\beta}{2}}}{\frac{\beta}{2}-1}\right] \right) \end{equation} where $J_\beta$ is the Bessel function. The solutions match those given in Solution to a second order ordinary differential equation .

Is there a way to derive the solutions using some different method?

Answer 1

Note, that integrating our ODE $q$ times and solving the resulting integral equation by successive iterations we end up with the following solution in terms of infinite series: \begin{equation} r_t = \sum\limits_{\eta=0}^{q-1} r^{(\eta)}_0 \sum\limits_{p=0}^\infty \frac{1}{((q-1)!)^p} I^{(t,t_ 0)}_{\vec{q}_\eta,p} \end{equation} where $\vec{q}_\eta := (q-1,q-1,\cdots,q-1,\eta)$ where the number $q-1$ is repeated $p$ times. The initial conditions of the ODE are \begin{equation} \frac{d^\eta r_t }{ d t^\eta}(t= t_0) = r^{(\eta)}_0 \end{equation} for $\eta=0,\cdots,q-1$ and the quantities $I^{(t,t_ 0)}_{\vec{q}_\eta,p}$ are multivariable integrals over a simplex, defined in Yet another multivariable integral over a simplex . For simplicity we only consider the last ($\eta=q-1$ ) term only. Using the closed form solution for the multivariable integral we easily arrive at the following special solution: \begin{eqnarray} &&\sum\limits_{l=0}^{q-1} \binom{q-1}{l} (-1)^l \\&& F_{0,q-1}\left[\begin{array}{c} \left\{\right\} \\ \left\{ 1 - \frac{l-\eta}{q-\beta}\right\}_{\stackrel{\eta=0}{\eta\neq l}}^{q-1}\end{array}; - \frac{t^{q-\beta}}{(q-\beta)^q}\right] F_{0,q-1}\left[\begin{array}{c} \left\{\right\} \\ \left\{ 1 + \frac{l+\eta}{q-\beta}\right\}_{\stackrel{\eta=0}{\eta\neq l}}^{q-1}\end{array}; (-1)^q \frac{t_0^{q-\beta}}{(q-\beta)^q}\right] t^{q-1-l} t_0^l \end{eqnarray} We will further simplify the result soon.