Internal problem ID [6847]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 115.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 57
dsolve(8*x^2*(1+2*x^2)*diff(y(x),x$2)+2*x*(5+34*x^2)*diff(y(x),x)-(1-30*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \LegendreP \left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )}{x^{\frac {1}{8}} \sqrt {2 x^{2}+1}}+\frac {c_{2} \LegendreQ \left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )}{x^{\frac {1}{8}} \sqrt {2 x^{2}+1}} \]
✓ Solution by Mathematica
Time used: 10.065 (sec). Leaf size: 54
DSolve[8*x^2*(1+2*x^2)*y''[x]+2*x*(5+34*x^2)*y'[x]-(1-30*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {3 c_1 x^{3/4}-4 c_2 \, _2F_1\left (-\frac {3}{8},\frac {1}{2};\frac {5}{8};-2 x^2\right )}{3 \sqrt {x} \sqrt {2 x^2+1}} \\ \end{align*}