Internal problem ID [9459]
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]]
\[ \boxed {x y^{\prime \prime }-2 \left (x^{2}-a \right ) y^{\prime }+2 n x y=0} \]
✓ Solution by Maple
Time used: 0.079 (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)
\[ y \left (x \right ) = c_{1} \operatorname {KummerM}\left (-\frac {n}{2}, \frac {1}{2}+a , x^{2}\right )+c_{2} \operatorname {KummerU}\left (-\frac {n}{2}, \frac {1}{2}+a , x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.13 (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]
\[ y(x)\to c_1 \operatorname {Hypergeometric1F1}\left (-\frac {n}{2},a+\frac {1}{2},x^2\right )+i^{1-2 a} c_2 x^{1-2 a} \operatorname {Hypergeometric1F1}\left (-a-\frac {n}{2}+\frac {1}{2},\frac {3}{2}-a,x^2\right ) \]