Internal problem ID [8855]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1275.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime } x^{2}+4 x y^{\prime }+\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.044 (sec). Leaf size: 55
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+(-x^2+2*(1-m+2*l)*x-m^2+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \WhittakerM \left (l -\frac {m}{2}+\frac {1}{2}, \frac {\sqrt {m +1}\, \sqrt {m -1}}{2}, x\right )}{\sqrt {x}}+\frac {c_{2} \WhittakerW \left (l -\frac {m}{2}+\frac {1}{2}, \frac {\sqrt {m +1}\, \sqrt {m -1}}{2}, x\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 97
DSolve[(1 - m^2 + 2*(1 + 2*l - m)*x - x^2)*y[x] + 4*x*y'[x] + 4*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x/2} x^{\frac {\sqrt {m^2-1}}{2}} \left (c_1 \text {HypergeometricU}\left (\frac {1}{2} \left (-2 l+\sqrt {m^2-1}+m\right ),\sqrt {m^2-1}+1,x\right )+c_2 L_{l-\frac {m}{2}-\frac {\sqrt {m^2-1}}{2}}^{\sqrt {m^2-1}}(x)\right ) \\ \end{align*}