60.2.245 problem 821

Internal problem ID [10832]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 821
Date solved : Monday, January 27, 2025 at 10:11:11 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {y \left (y x +1\right )}{x \left (-y x -1+y^{4} x^{3}\right )} \end{align*}

✓ Solution by Maple

Time used: 0.008 (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)
 

\[ -\frac {1}{2 x^{2} y^{2}}-\frac {1}{3 y^{3} x^{3}}-y+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.177 (sec). Leaf size: 1993

DSolve[D[y[x],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*}