Internal problem ID [40]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.4. Separable equations. Page 43
Problem number: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (x -1\right ) y^{5}}{x^{2} \left (-y+2 y^{3}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 2094
dsolve(diff(y(x),x) = (-1+x)*y(x)^5/x^2/(-y(x)+2*y(x)^3),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {4^{\frac {1}{3}} \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}{6 x \ln \relax (x )+6 x c_{1}+6}+\frac {2 x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \relax (x )+x c_{1}+1\right ) \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}-\frac {2 x}{3 \left (x \ln \relax (x )+x c_{1}+1\right )} \\ y \relax (x ) = -\frac {4^{\frac {1}{3}} \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}{12 \left (x \ln \relax (x )+x c_{1}+1\right )}-\frac {x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \relax (x )+x c_{1}+1\right ) \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}-\frac {2 x}{3 \left (x \ln \relax (x )+x c_{1}+1\right )}-\frac {i \sqrt {3}\, \left (\frac {4^{\frac {1}{3}} \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}{6 x \ln \relax (x )+6 x c_{1}+6}-\frac {2 x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \relax (x )+x c_{1}+1\right ) \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {4^{\frac {1}{3}} \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}{12 \left (x \ln \relax (x )+x c_{1}+1\right )}-\frac {x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \relax (x )+x c_{1}+1\right ) \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}-\frac {2 x}{3 \left (x \ln \relax (x )+x c_{1}+1\right )}+\frac {i \sqrt {3}\, \left (\frac {4^{\frac {1}{3}} \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}{6 x \ln \relax (x )+6 x c_{1}+6}-\frac {2 x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \relax (x )+x c_{1}+1\right ) \left (x \left (9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, \ln \relax (x ) x +3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}\, c_{1} x +18 x \ln \relax (x )+18 x c_{1}-16 x^{2}+3 \sqrt {9 \ln \relax (x )^{2} x^{2}+18 \ln \relax (x ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \relax (x )+18 x c_{1}-32 x^{2}+9}+9\right )\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 22.522 (sec). Leaf size: 842
DSolve[y'[x] == (-1+x)*y[x]^5/x^2/(-y[x]+2*y[x]^3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\frac {8 \sqrt [3]{2} x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}+4 x}{6 (x \log (x)+c_1 x+1)} \\ y(x)\to \frac {\frac {8 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}-8 x}{12 (x \log (x)+c_1 x+1)} \\ y(x)\to \frac {\frac {8 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}-8 x}{12 (x \log (x)+c_1 x+1)} \\ y(x)\to 0 \\ \end{align*}