Internal problem ID [7240]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 520.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {12 x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.094 (sec). Leaf size: 51
dsolve(12*x^2*(1+x)*diff(y(x),x$2)+x*(11+35*x+3*x^2)*diff(y(x),x)-(1-10*x-5*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-\frac {x}{4}} \HeunC \left (\frac {1}{4}, -\frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right )}{x^{\frac {1}{4}} \left (x +1\right )^{\frac {3}{4}}}+\frac {c_{2} {\mathrm e}^{-\frac {x}{4}} \HeunC \left (\frac {1}{4}, \frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right ) x^{\frac {1}{3}}}{\left (x +1\right )^{\frac {3}{4}}} \]
✓ Solution by Mathematica
Time used: 10.181 (sec). Leaf size: 61
DSolve[12*x^2*(1+x)*y''[x]+x*(11+35*x+3*x^2)*y'[x]-(1-10*x-5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-x/4} \left (c_2 \int _1^x\frac {e^{\frac {K[1]}{4}}}{K[1]^{5/12} \sqrt [4]{K[1]+1}}dK[1]+c_1\right )}{\sqrt [4]{x} (x+1)^{3/4}} \\ \end{align*}