59.1.491 problem 507

Internal problem ID [9663]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 507
Date solved : Monday, January 27, 2025 at 06:12:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (4+x \right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 53

dsolve((4+x)*diff(y(x),x$2)+(2+x)*diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
 

\[ y = x c_{2} {\mathrm e}^{-x -4} \left (x +4\right )^{3} \operatorname {Ei}_{1}\left (-x -4\right )+c_{1} {\mathrm e}^{-x} x \left (x +4\right )^{3}+c_{2} \left (x^{3}+9 x^{2}+22 x +6\right ) \]

✓ Solution by Mathematica

Time used: 0.354 (sec). Leaf size: 93

DSolve[(4+x)*D[y[x],{x,2}]+(2+x)*D[y[x],x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \exp \left (\int _1^x-\frac {K[1]}{2 (K[1]+4)}dK[1]-\frac {1}{2} \int _1^x\frac {K[2]+2}{K[2]+4}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}-\frac {K[1]}{2 (K[1]+4)}dK[1]\right )}{K[3]^2}dK[3]+c_1\right ) \]