Internal problem ID [7748]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 168.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {3 \left (x^{2}-4\right ) y^{\prime }+y^{2}-y x -3=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 140
dsolve(3*(x^2-4)*diff(y(x),x) + y(x)^2 - x*y(x) - 3=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {3 \left (x +2\right ) \left (\HeunC \left (0, \frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right ) c_{1}-\frac {\left (-\frac {x}{4}-\frac {1}{2}\right )^{\frac {4}{3}} \HeunC \left (0, -\frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right )}{3}\right )}{4 \left (x -\frac {5}{4}\right ) \left (x +2\right ) c_{1} \HeunC \left (0, \frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right )-\left (-\frac {x}{4}-\frac {1}{2}\right )^{\frac {4}{3}} \left (x +2\right ) \HeunC \left (0, -\frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right )+12 \left (\HeunCPrime \left (0, \frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right ) c_{1}-\frac {\left (-\frac {x}{4}-\frac {1}{2}\right )^{\frac {4}{3}} \HeunCPrime \left (0, -\frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right )}{3}\right ) \left (x -2\right )} \]
✓ Solution by Mathematica
Time used: 0.391 (sec). Leaf size: 135
DSolve[3*(x^2-4)*y'[x] + y[x]^2 - x*y[x] - 3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 c_1 x P_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )+3 c_1 P_{\frac {5}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )-2 x Q_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )+3 Q_{\frac {5}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )}{Q_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )+c_1 P_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )} \\ y(x)\to \frac {3 P_{\frac {5}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )}{P_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )}-2 x \\ \end{align*}