59.1.240 problem 243

Internal problem ID [9412]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 243
Date solved : Monday, January 27, 2025 at 06:02:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 27

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

\[ y = \frac {c_{2} \left (x +2\right ) \ln \left (x +1\right )+c_{1} x +2 c_{1} +4 c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.484 (sec). Leaf size: 87

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

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