Internal problem ID [8401]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 821.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (y x +1\right )}{x \left (-y x -1+y^{4} x^{3}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 27
dsolve(diff(y(x),x) = 1/x*y(x)*(x*y(x)+1)/(-x*y(x)-1+y(x)^4*x^3),y(x), singsol=all)
\[ -\frac {1}{2 x^{2} y \relax (x )^{2}}-\frac {1}{3 y \relax (x )^{3} x^{3}}-y \relax (x )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.157 (sec). Leaf size: 1993
DSolve[y'[x] == (y[x]*(1 + x*y[x]))/(x*(-1 - x*y[x] + x^3*y[x]^4)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {-8-3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}-\frac {-\frac {4}{x^2}+c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {-8-3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}-\frac {-\frac {4}{x^2}+c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {-8-3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {-\frac {4}{x^2}+c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {-8-3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {-\frac {4}{x^2}+c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}{6 x^3}+\frac {8+3 c_1 x}{3 \sqrt [3]{36 c_1{}^2 x^6+27 x^5+\sqrt {x^9 \left (216 (-1+6 c_1) c_1{}^3 x^3+216 c_1{}^2 x^2-9 (-81+512 c_1) x-4096\right )}}}+\frac {c_1{}^2}{4}}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ \end{align*}