Internal problem ID [6868]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 136.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (x +2\right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-x \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (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)
\[ y \relax (x ) = c_{1} \sqrt {x \left (x +2\right )}+\frac {c_{2} \sqrt {x}\, \left (-\left (x +2\right ) \arctanh \left (\frac {\sqrt {x +2}\, \sqrt {2}}{2}\right )+\sqrt {x +2}\, \sqrt {2}\right )}{\sqrt {x +2}} \]
✓ Solution by Mathematica
Time used: 0.109 (sec). Leaf size: 65
DSolve[2*x^2*(2+x)*y''[x]+x^2*y'[x]+(1-x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {x} \left (2 \left (c_1 \sqrt {x+2}+c_2\right )-\sqrt {2} c_2 \sqrt {x+2} \tanh ^{-1}\left (\frac {\sqrt {x+2}}{\sqrt {2}}\right )\right )}{2 \sqrt [4]{2}} \\ \end{align*}