Internal problem ID [6959]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 228.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 47
dsolve((1+2*x^2)*diff(y(x),x$2)+7*x*diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \LegendreP \left (\frac {1}{4}, \frac {3}{4}, i \sqrt {2}\, x \right )}{\left (2 x^{2}+1\right )^{\frac {3}{8}}}+\frac {c_{2} \LegendreQ \left (\frac {1}{4}, \frac {3}{4}, i \sqrt {2}\, x \right )}{\left (2 x^{2}+1\right )^{\frac {3}{8}}} \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 66
DSolve[(1+2*x^2)*y''[x]+7*x*y'[x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 Q_{\frac {1}{4}}^{\frac {3}{4}}\left (i \sqrt {2} x\right )}{\left (2 x^2+1\right )^{3/8}}+\frac {2 i \sqrt [4]{2} c_1 x}{\left (2 x^2+1\right )^{3/4} \text {Gamma}\left (\frac {1}{4}\right )} \\ \end{align*}