I want to find all spherical standing waves in $\mathbb{R}^3$, i.e. all solutions of homogeneous wave equation in $\mathbb{R}^3$ of the form $u(x,t) = w(|x|) \sin(\omega t + \phi)$.
I noted that, if such $u(x,t)$ solves the wave equation, we must have:
\begin{equation} u_{tt} - \Delta u = 0 \\ -\omega^2 w(|x|) \sin(\omega t + \phi) - (w''(|x|) + 2 w'(|x|) / |x|) \sin(\omega t + \phi) = 0 \end{equation} So, in particular, any such $w$ must solve this ODE for $x > 0$: \begin{equation} \omega^2 w(x) + \frac{2w'(x)}{x} + w''(x) = 0 \end{equation} However, I don't know how to solve this ODE - can anyone help me with that? Or are there other ways to find all spherical standing waves in $\mathbb{R}^3$?
$$\omega^2 w(x) + \frac{2w'(x)}{x} + w''(x) = 0$$ Substitute $u=\omega x$: $$ w(u) + \frac{2w'(u)}{u} + w''(u) = 0$$ $$ uw(u) +{2w'(u)} + uw''(u) = 0$$ $$ uw(u) +( uw(u))'' = 0$$ This should be integrable since it's a second order linear differential equation with constant coefficients. $$y''(u)+y(u)=0$$ $$y(u)=c_1 \cos u +c_2 \sin u$$ where $y=uw(u)$. $$w(x)=\dfrac 1 {\omega x}\left(c_1 \cos (\omega x) +c_2 \sin (\omega x)\right)$$