Internal problem ID [6990]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 259.
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 (x +n \right ) y^{\prime }+\left (n +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 42
dsolve(x*diff(y(x),x$2)+(x+n)*diff(y(x),x)+(n+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-x} \hypergeom \left (\left [-1\right ], \relax [n ], x\right )+c_{2} {\mathrm e}^{-x} x^{-n +1} \hypergeom \left (\left [-n \right ], \left [-n +2\right ], x\right ) \]
✓ Solution by Mathematica
Time used: 0.639 (sec). Leaf size: 48
DSolve[x*y''[x]+(x+n)*y'[x]+(n+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} (n-x) \left (c_2 \int _1^x\frac {e^{K[1]} K[1]^{-n}}{(n-K[1])^2}dK[1]+c_1\right ) \\ \end{align*}