Internal problem ID [7338]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 618.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.112 (sec). Leaf size: 76
dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(7-2*x^2)*diff(y(x),x)+12*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} x^{6}}{\left (x^{2}+1\right )^{\frac {7}{2}}}+\frac {c_{2} x^{2} \left (8 \sqrt {x^{2}+1}\, x^{4}-15 \arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right ) x^{4}-9 \sqrt {x^{2}+1}\, x^{2}-2 \sqrt {x^{2}+1}\right )}{\left (x^{2}+1\right )^{\frac {7}{2}}} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 88
DSolve[x^2*(1+x^2)*y''[x]-x*(7-2*x^2)*y'[x]+12*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 c_2 \sqrt {x^2+1} x^2+8 x^6 \left (c_2 \sqrt {x^2+1}+c_1\right )-15 c_2 x^6 \tanh ^{-1}\left (\sqrt {x^2+1}\right )-9 c_2 \sqrt {x^2+1} x^4}{8 \left (x^2+1\right )^{7/2}} \\ \end{align*}