59.1.134 problem 136

Internal problem ID [9306]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 136
Date solved : Monday, January 27, 2025 at 06:01:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 50

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

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

✓ Solution by Mathematica

Time used: 0.508 (sec). Leaf size: 92

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

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