59.1.248 problem 251

Internal problem ID [9420]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 251
Date solved : Monday, January 27, 2025 at 06:02:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x \left (x +2\right ) y^{\prime }+2 y \left (1+x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 28

dsolve(x^2*diff(y(x),x$2)+2*x*(2+x)*diff(y(x),x)+2*(1+x)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.239 (sec). Leaf size: 33

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

\[ y(x)\to \frac {c_2 \int _1^x\frac {e^{-2 K[1]}}{K[1]^2}dK[1]+c_1}{x} \]