Internal problem ID [6792]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 60.
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 }-9 x y^{\prime }-6 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 47
dsolve((1+2*x^2)*diff(y(x),x$2)-9*x*diff(y(x),x)-6*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (2 x^{2}+1\right )^{\frac {13}{8}} \LegendreP \left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right )+c_{2} \left (2 x^{2}+1\right )^{\frac {13}{8}} \LegendreQ \left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 69
DSolve[(1+2*x^2)*y''[x]-9*x*y'[x]-6*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \left (2 x^2+1\right )^{13/8} Q_{\frac {11}{4}}^{\frac {13}{4}}\left (i \sqrt {2} x\right )-\frac {48 \sqrt [4]{2} c_1 \left (3 x^6+5 x^4+3 x^2+1\right )}{\text {Gamma}\left (-\frac {5}{4}\right )} \\ \end{align*}