Internal problem ID [8696]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1116.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+\left (a x +b +n \right ) y^{\prime }+n a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 35
dsolve(x*diff(diff(y(x),x),x)+(a*x+b+n)*diff(y(x),x)+n*a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-a x} \KummerM \left (b , b +n , a x \right )+c_{2} {\mathrm e}^{-a x} \KummerU \left (b , b +n , a x \right ) \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 38
DSolve[a*n*y[x] + (b + n + a*x)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-a x} (c_1 \text {HypergeometricU}(b,b+n,a x)+c_2 L_{-b}^{b+n-1}(a x)) \\ \end{align*}