Internal problem ID [8046]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 570.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 x^{2} \left (x +2\right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1-x \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 71
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)
\[ y \left (x \right ) = c_{1} \sqrt {x^{2}+2 x}-\frac {c_{2} \sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x +2}}{2}\right ) x -\sqrt {2}\, \sqrt {x +2}+2 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x +2}}{2}\right )\right ) \sqrt {x \left (x +2\right )}}{2 \left (x +2\right )} \]
✓ Solution by Mathematica
Time used: 0.1 (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]
\[ y(x)\to \frac {\sqrt {x} \left (2 \left (c_1 \sqrt {x+2}+c_2\right )-\sqrt {2} c_2 \sqrt {x+2} \text {arctanh}\left (\frac {\sqrt {x+2}}{\sqrt {2}}\right )\right )}{2 \sqrt [4]{2}} \]