Internal problem ID [6862]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 130.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.601 (sec). Leaf size: 52
dsolve(x^2*(1+2*x)*diff(y(x),x$2)+x*(5+14*x+3*x^2)*diff(y(x),x)+(4+18*x+12*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-\frac {3 x}{2}} \HeunC \left (-\frac {3}{4}, -\frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right )}{x^{2} \left (2 x +1\right )^{\frac {1}{4}}}+\frac {c_{2} {\mathrm e}^{-\frac {3 x}{2}} \HeunC \left (-\frac {3}{4}, \frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 11.215 (sec). Leaf size: 61
DSolve[x^2*(1+2*x)*y''[x]+x*(5+14*x+3*x^2)*y'[x]+(4+18*x+12*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-3 x/2} \left (c_2 \int _1^x\frac {e^{\frac {3 K[1]}{2}}}{K[1] (2 K[1]+1)^{3/4}}dK[1]+c_1\right )}{x^2 \sqrt [4]{2 x+1}} \\ \end{align*}