Internal problem ID [6820]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 88.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {18 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.046 (sec). Leaf size: 41
dsolve(18*x^2*(1+x)*diff(y(x),x$2)+3*x*(5+11*x+x^2)*diff(y(x),x)-(1-2*x-5*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-\frac {x}{6}} \HeunC \left (\frac {1}{6}, -\frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right )}{x^{\frac {1}{6}}}+c_{2} {\mathrm e}^{-\frac {x}{6}} \HeunC \left (\frac {1}{6}, \frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right ) x^{\frac {1}{3}} \]
✓ Solution by Mathematica
Time used: 1.154 (sec). Leaf size: 60
DSolve[18*x^2*(1+x)*y''[x]+3*x*(5+11*x+x^2)*y'[x]-(1-2*x-5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-x/6} \left (c_2 \int _1^x\frac {e^{\frac {K[1]}{6}}}{\sqrt {K[1]} (K[1]+1)^{7/6}}dK[1]+c_1\right )}{\sqrt [6]{\frac {x}{x+1}}} \\ \end{align*}