59.1.256 problem 259

Internal problem ID [9428]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 259
Date solved : Monday, January 27, 2025 at 06:02:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.022 (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)
 

\[ y = \frac {\left (c_{2} x^{-n +1} \operatorname {hypergeom}\left (\left [-n \right ], \left [-n +2\right ], x\right ) n +\left (-x +n \right ) c_{1} \right ) {\mathrm e}^{-x}}{n} \]

✓ Solution by Mathematica

Time used: 0.355 (sec). Leaf size: 77

DSolve[x*D[y[x],{x,2}]+(x+n)*D[y[x],x]+(n+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{\int \frac {-n+x-1}{n-x} \, dx} \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {n^2-2 K[1] n+(K[1]-2) K[1]}{(n-K[1]) K[1]}dK[1]\right )dK[2]+c_1\right ) \]