59.1.542 problem 558

Internal problem ID [9714]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 558
Date solved : Monday, January 27, 2025 at 06:13:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x +2\right ) y^{\prime \prime }+5 x^{2} y^{\prime }+y \left (1+x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 39

dsolve(2*x^2*(2+x)*diff(y(x),x$2)+5*x^2*diff(y(x),x)+(1+x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\sqrt {x}\, \left (\sqrt {2}\, \sqrt {x +2}\, c_{2} -2 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x +2}}{2}\right ) c_{2} +c_{1} \right )}{\left (x +2\right )^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.392 (sec). Leaf size: 83

DSolve[2*x^2*(2+x)*D[y[x],{x,2}]+5*x^2*D[y[x],x]+(1+x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {K[1]+4}{4 K[1]^2+8 K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+4}{4 K[1]^2+8 K[1]}dK[1]\right )dK[2]+c_1\right )}{(x+2)^{5/4}} \]