Internal problem ID [6842]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 110.
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}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.359 (sec). Leaf size: 31
dsolve(2*x^2*(1+x^2)*diff(y(x),x$2)+x*(3+8*x^2)*diff(y(x),x)-(3-4*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \hypergeom \left (\left [1, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )+\frac {c_{2}}{x^{\frac {3}{2}} \left (x^{2}+1\right )^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 10.054 (sec). Leaf size: 49
DSolve[2*x^2*(1+x^2)*y''[x]+x*(3+8*x^2)*y'[x]-(3-4*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {c_2 \, _2F_1\left (\frac {1}{2},1;\frac {5}{4};-x^2\right )}{x}+\frac {c_1}{x^{3/2} \sqrt [4]{x^2+1}}+\frac {c_2}{x} \\ \end{align*}