Second order ODE with non-constant coefficients

Question

I'd like to know how to solve the differential equation: $u_{tt}+\frac{1-2s}{t}u_t-u=0$ where s is a constant. Let $u(0)=1$ and $\lim_{t \to 0+} t^{1-2s}u_t = c$. I'm studying a fractional laplacian equation and this ODE arose after applying the Caffarelli-Silvestre extension (https://arxiv.org/pdf/math/0608640.pdf). I have only learnt how to solve second order linear ODEs with constant coefficients, so I wasn't sure how to go about solving this equation.

Answer 1

Let $u=t^{s} v$. Therefore we have $u' = t^{s}v' + st^{s-1}v$ and $u'' = t^{s}v'' + 2s t^{s-1}v' + s(s-1)t^{s-2}v$. Substituting into the ODE yields \begin{align} t^{s}v'' + 2s t^{s-1}v' + s(s-1)t^{s-2}v + \frac{1-2s}{t}(t^{s}v' + st^{s-1}v) - t^{s} v = & \, 0 \\ \implies t^2v'' + 2s tv' + s(s-1)v + \frac{1-2s}{t}(t^{2}v' + stv) - t^2v = & \, 0\\ \implies t^2v'' + tv' -(s^2+t^2)v = & \, 0. \end{align} This is the modified Bessel equation. To get to the classic Bessel equation we make the transformation $t\to i x$. Therefore $\frac{d}{dt}=-i \frac{d}{dx}$ and thus $$ x^2 v'' + x v' + (x^2-s^2)v=0.$$ This is Bessel's equation https://en.wikipedia.org/wiki/Bessel_function.

Edit:

In response to the comment on initial conditions I am expanding my answer a bit. Let us consider the solution $$ u(t) =t^s\left ( C_1 I_s(t) + C_2 K_s(t) \right ) $$ where $I_s$ and $K_s$ are modified Bessel functions. Assuming $s> 0$ (or $\Re (s) > 0$ for the complex case) the condition $u(0)=1$ leads to $$ 1 = \lim_{t\to 0^+} t^s\left ( C_1 I_s(t) + C_2 K_s(t) \right ) = C_2 \lim_{t\to 0^+}t^s K_s(t).$$ This still works as $K_s(t)=O(t^{-s})$ as $t\to 0$. I think (but you should check) that we get $$ \lim_{t\to 0^+} t^s K_s(t) = 2^{s-1}\Gamma (s). $$ Thus we find that $C_2 = \frac{1}{\Gamma(s)} 2^{1-s}$. The case $s=0$ needs to be considered separately. I will leave it to you to solve for the other condition.

Note: I checked the other condition and I think both can only be satisfied if $0 \le s \le 1$. If this is an unacceptable limitation then you may want to check both my work, and your work in deriving the equation and IC's. A physical interpretation of what $s$ is may help.