Internal problem ID [8704]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1124.
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 \left (x^{2}-a \right ) y^{\prime }+2 n x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 29
dsolve(x*diff(diff(y(x),x),x)-2*(x^2-a)*diff(y(x),x)+2*n*x*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \KummerM \left (-\frac {n}{2}, \frac {1}{2}+a , x^{2}\right )+c_{2} \KummerU \left (-\frac {n}{2}, \frac {1}{2}+a , x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 65
DSolve[2*n*x*y[x] - 2*(-a + x^2)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _1F_1\left (-\frac {n}{2};a+\frac {1}{2};x^2\right )+i^{1-2 a} c_2 x^{1-2 a} \, _1F_1\left (-a-\frac {n}{2}+\frac {1}{2};\frac {3}{2}-a;x^2\right ) \\ \end{align*}