Internal problem ID [8751]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1171.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+2 x y^{\prime }+\left (l \,x^{2}+a x -n \left (n +1\right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 51
dsolve(x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+(l*x^2+a*x-n*(n+1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \WhittakerM \left (-\frac {i a}{2 \sqrt {l}}, n +\frac {1}{2}, 2 i \sqrt {l}\, x \right )}{x}+\frac {c_{2} \WhittakerW \left (-\frac {i a}{2 \sqrt {l}}, n +\frac {1}{2}, 2 i \sqrt {l}\, x \right )}{x} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 92
DSolve[(-(n*(1 + n)) + a*x + l*x^2)*y[x] + 2*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-i \sqrt {l} x} x^n \left (c_1 \text {HypergeometricU}\left (\frac {i a}{2 \sqrt {l}}+n+1,2 n+2,2 i \sqrt {l} x\right )+c_2 L_{-\frac {i a}{2 \sqrt {l}}-n-1}^{2 n+1}\left (2 i \sqrt {l} x\right )\right ) \\ \end{align*}