Internal problem ID [6845]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 113.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 33
dsolve(9*x^2*(1+x^2)*diff(y(x),x$2)+3*x*(3+13*x^2)*diff(y(x),x)-(1-25*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{x^{\frac {1}{3}} \left (x^{2}+1\right )^{\frac {2}{3}}}+c_{2} x^{\frac {1}{3}} \hypergeom \left (\left [1, 1\right ], \left [\frac {4}{3}\right ], -x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.67 (sec). Leaf size: 115
DSolve[9*x^2*(1+x^2)*y''[x]+3*x*(3+13*x^2)*y'[x]-(1-25*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 \left (-2 \log \left (\sqrt [3]{x^2+1}-x^{2/3}\right )+\log \left (x^{4/3}+\left (x^2+1\right )^{2/3}+\sqrt [3]{x^2+1} x^{2/3}\right )+2 \sqrt {3} \cot ^{-1}\left (\frac {\frac {2 \sqrt [3]{x^2+1}}{x^{2/3}}+1}{\sqrt {3}}\right )\right )+4 c_1}{4 \sqrt [3]{x} \left (x^2+1\right )^{2/3}} \\ \end{align*}