Internal problem ID [8672]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1092.
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 +\left (x +a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 29
dsolve(x*diff(diff(y(x),x),x)+(x+a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \WhittakerM \left (-\frac {i a}{2}, \frac {1}{2}, 2 i x \right )+c_{2} \WhittakerW \left (-\frac {i a}{2}, \frac {1}{2}, 2 i x \right ) \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 53
DSolve[(a + x)*y[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-i x} x \left (c_1 \text {HypergeometricU}\left (1+\frac {i a}{2},2,2 i x\right )+c_2 \, _1F_1\left (\frac {i a}{2}+1;2;2 i x\right )\right ) \\ \end{align*}