3.14 problem 15

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: 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 \left (x \right ) = \frac {4^{\frac {1}{3}} {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}{6 x \ln \left (x \right )+6 c_{1} x +6}+\frac {2 x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \left (x \right )+c_{1} x +1\right ) {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}-\frac {2 x}{3 \left (x \ln \left (x \right )+c_{1} x +1\right )} y \left (x \right ) = -\frac {4^{\frac {1}{3}} {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}{12 \left (x \ln \left (x \right )+c_{1} x +1\right )}-\frac {x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \left (x \right )+c_{1} x +1\right ) {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}-\frac {2 x}{3 \left (x \ln \left (x \right )+c_{1} x +1\right )}-\frac {i \sqrt {3}\, \left (\frac {4^{\frac {1}{3}} {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}{6 x \ln \left (x \right )+6 c_{1} x +6}-\frac {2 x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \left (x \right )+c_{1} x +1\right ) {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {4^{\frac {1}{3}} {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}{12 \left (x \ln \left (x \right )+c_{1} x +1\right )}-\frac {x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \left (x \right )+c_{1} x +1\right ) {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}-\frac {2 x}{3 \left (x \ln \left (x \right )+c_{1} x +1\right )}+\frac {i \sqrt {3}\, \left (\frac {4^{\frac {1}{3}} {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}{6 x \ln \left (x \right )+6 c_{1} x +6}-\frac {2 x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \left (x \right )+c_{1} x +1\right ) {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}\right )}{2} \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*}