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]
\[ \boxed {y^{\prime }-\frac {\left (x -1\right ) y^{5}}{x^{2} \left (-y+2 y^{3}\right )}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 844
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 \left (x \right ) &= \frac {8 x^{2} 2^{\frac {1}{3}}-4 x \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {2}{3}}}{\left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}} \left (6 c_{1} x +6 x \ln \left (x \right )+6\right )} \\ y \left (x \right ) &= -\frac {8 x \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}}-8 x^{2} \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}+2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {2}{3}}}{\left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}} \left (12 c_{1} x +12 x \ln \left (x \right )+12\right )} \\ y \left (x \right ) &= \frac {-8 x \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}}-8 x^{2} \left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}}+2^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {2}{3}}}{\left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}} \left (12 c_{1} x +12 x \ln \left (x \right )+12\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.626 (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*}