Here is a differential equation that I encountered while solving a quantum mechanics problem :
$$ \frac{d}{dr} \left(r^2 \frac{dR}{dr} \right) - k^2r^2R = l(l+1)R, $$ where $R = R(r) \ \& \ k \ , l \ $ are constants.
This can be reduced to : $$ r^2 \frac{d^2R}{dr^2} + 2r \frac{dR}{dr} - (k^2r^2 + l(l+1) )R = 0. $$
The solution to this equation is : $$ A \ J_{-l(l+1)} (-ikr) + B \ Y_{-l(l+1)} (-ikr). $$
But in another approach a new variable $u(r)$ was defined as $ u(r) = r \ R(r)$.
Hence the resulting differential equation is :
$$ \frac{d^2u}{dr^2} - (l(l+1)/r^2) u - k^2 u=0. $$
The solution to this equation is : $$ u(r) = A \ J_{l+ (1/2)}(-ikr) r^{1/2} + B \ Y_{l+(1/2)}(-ikr)r^{1/2}. $$
which gives $$ R(r) = u(r)/r =A \ J_{l+ (1/2)}(-ikr) r^{-1/2} + B \ Y_{l+(1/2)}(-ikr)r^{-1/2}. $$
My query is : Why are the two solutions not equivalent ?
$A \ J_{-l(l+1)} (-ikr) + B \ Y_{-l(l+1)} (-ikr)$ is solution of :
$$r^2 \frac{d^2R}{dr^2} + r \frac{dR}{dr} - (k^2r^2 + l^2(l+1)^2 )R = 0$$ but is NOT solution of : $$r^2 \frac{d^2R}{dr^2} + 2r \frac{dR}{dr} - (k^2r^2 + l(l+1) )R = 0$$ which solution is : $$A \ j_{l} (-ikr) + B \ y_{l} (-ikr)$$ where $j$ and $y$ denote the spherical Bessel functions. Then, use the relationships between the spherical Bessel functions $j$ , $y$ and the Bessel functions $J$, $Y$ respectively.
See :http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html