Internal problem ID [7261]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 541.
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}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.145 (sec). Leaf size: 39
dsolve(2*x^2*(2+x^2)*diff(y(x),x$2)+x*(4+7*x^2)*diff(y(x),x)-(1-3*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{\left (x^{2}+2\right )^{\frac {1}{4}} \sqrt {x}}+\frac {c_{2} \LegendreQ \left (-\frac {1}{4}, \frac {1}{4}, \frac {i \sqrt {2}\, x}{2}\right )}{\left (x^{2}+2\right )^{\frac {1}{8}} \sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 68
DSolve[2*x^2*(2+x^2)*y''[x]+x*(4+7*x^2)*y'[x]-(1-3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 \sqrt [8]{x^2+2} \operatorname {Gamma}\left (\frac {3}{4}\right ) Q_{-\frac {1}{4}}^{\frac {1}{4}}\left (\frac {i x}{\sqrt {2}}\right )+2^{3/8} c_1}{\sqrt {x} \sqrt [4]{x^2+2} \operatorname {Gamma}\left (\frac {3}{4}\right )} \\ \end{align*}