Internal problem ID [7962]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 485.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 85
dsolve(x^2*(2+x^2)*diff(y(x),x$2)+2*x*(x^2+5)*diff(y(x),x)+2*(3-x^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+8\right )}{x}-\frac {c_{2} \sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right ) x^{4}-\sqrt {2}\, \sqrt {x^{2}+2}\, x^{2}+8 \,\operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right ) x^{2}+4 \sqrt {2}\, \sqrt {x^{2}+2}\right )}{64 x^{3}} \]
✓ Solution by Mathematica
Time used: 0.135 (sec). Leaf size: 88
DSolve[x^2*(2+x^2)*y''[x]+2*x*(x^2+5)*y'[x]+2*(3-x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {-\sqrt {2} c_2 \left (x^2+8\right ) x^2 \text {arctanh}\left (\frac {\sqrt {x^2+2}}{\sqrt {2}}\right )+64 c_1 x^4+2 x^2 \left (c_2 \sqrt {x^2+2}+256 c_1\right )-8 c_2 \sqrt {x^2+2}}{64 x^3} \]