Internal problem ID [7759]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 179.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y-3 x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 145
dsolve(3*x*(x^2-1)*diff(y(x),x) + x*y(x)^2 - (x^2+1)*y(x) - 3*x=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (35 c_{1} x^{4}-35 c_{1} x^{2}\right ) \hypergeom \left (\left [\frac {11}{6}, \frac {13}{6}\right ], \left [\frac {7}{3}\right ], x^{2}\right )}{8 x^{\frac {1}{3}} \left (\hypergeom \left (\left [\frac {5}{6}, \frac {7}{6}\right ], \left [\frac {4}{3}\right ], x^{2}\right ) x^{\frac {2}{3}} c_{1}+\hypergeom \left (\left [\frac {1}{2}, \frac {5}{6}\right ], \left [\frac {2}{3}\right ], x^{2}\right )\right )}+\frac {\left (40 c_{1} x^{2}-16 c_{1}\right ) \hypergeom \left (\left [\frac {5}{6}, \frac {7}{6}\right ], \left [\frac {4}{3}\right ], x^{2}\right )+\left (30 x^{\frac {10}{3}}-30 x^{\frac {4}{3}}\right ) \hypergeom \left (\left [\frac {3}{2}, \frac {11}{6}\right ], \left [\frac {5}{3}\right ], x^{2}\right )+24 \hypergeom \left (\left [\frac {1}{2}, \frac {5}{6}\right ], \left [\frac {2}{3}\right ], x^{2}\right ) x^{\frac {4}{3}}}{8 x^{\frac {1}{3}} \left (\hypergeom \left (\left [\frac {5}{6}, \frac {7}{6}\right ], \left [\frac {4}{3}\right ], x^{2}\right ) x^{\frac {2}{3}} c_{1}+\hypergeom \left (\left [\frac {1}{2}, \frac {5}{6}\right ], \left [\frac {2}{3}\right ], x^{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 3.166 (sec). Leaf size: 715
DSolve[3*x*(x^2-1)*y'[x] + x*y[x]^2 - (x^2+1)*y[x] - 3*x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\frac {6 \left (x^2-1\right ) \exp \left (-2 \int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\&,1\right ]dK[1]\right )}{\int _1^x\exp \left (-2 \int _1^{K[2]}\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\&,1\right ]dK[1]\right )dK[2]+c_1}+6 \left (x^2-1\right ) \text {Root}\left [\text {$\#$1}^4 \left (1296 x^{12}-5184 x^{10}+7776 x^8-5184 x^6+1296 x^4\right )+\text {$\#$1}^3 \left (-3456 x^{11}+12096 x^9-15552 x^7+8640 x^5-1728 x^3\right )+\text {$\#$1}^2 \left (3240 x^{10}-9504 x^8+9936 x^6-4320 x^4+648 x^2\right )+\text {$\#$1} \left (-1200 x^9+2736 x^7-2160 x^5+720 x^3-96 x\right )+125 x^8-164 x^6+70 x^4-20 x^2+5\&,1\right ]-5 x+\frac {1}{x}\right ) \\ y(x)\to \frac {1}{2} \left (6 \left (x^2-1\right ) \text {Root}\left [\text {$\#$1}^4 \left (1296 x^{12}-5184 x^{10}+7776 x^8-5184 x^6+1296 x^4\right )+\text {$\#$1}^3 \left (-3456 x^{11}+12096 x^9-15552 x^7+8640 x^5-1728 x^3\right )+\text {$\#$1}^2 \left (3240 x^{10}-9504 x^8+9936 x^6-4320 x^4+648 x^2\right )+\text {$\#$1} \left (-1200 x^9+2736 x^7-2160 x^5+720 x^3-96 x\right )+125 x^8-164 x^6+70 x^4-20 x^2+5\&,1\right ]-5 x+\frac {1}{x}\right ) \\ \end{align*}