Internal problem ID [40]
Internal file name [OUTPUT/40_Sunday_June_05_2022_01_33_58_AM_70028064/index.tex
]
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.
The type(s) of ODE detected by this program : "exact", "separable", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {\left (x -1\right ) y^{5}}{x^{2} \left (-y+2 y^{3}\right )}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y^{4} \left (x -1\right )}{x^{2} \left (2 y^{2}-1\right )} \end {align*}
Where \(f(x)=\frac {x -1}{x^{2}}\) and \(g(y)=\frac {y^{4}}{2 y^{2}-1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y^{4}}{2 y^{2}-1}} \,dy &= \frac {x -1}{x^{2}} \,d x \\ \int { \frac {1}{\frac {y^{4}}{2 y^{2}-1}} \,dy} &= \int {\frac {x -1}{x^{2}} \,d x} \\ \frac {-2 y^{2}+\frac {1}{3}}{y^{3}}&=\frac {1}{x}+\ln \left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \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 &= -\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 &= -\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*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \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 )} \\ \tag{2} y &= -\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} \\ \tag{3} y &= -\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*}
Verification of solutions
\[ y = \frac {4^{\frac {1}{3}} {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}{6 x \ln \left (x \right )+6 c_{1} x +6}+\frac {2 x^{2} 4^{\frac {2}{3}}}{3 \left (x \ln \left (x \right )+c_{1} x +1\right ) {\left (x \left (9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+3 \ln \left (x \right ) \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, x +3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}\, c_{1} x +18 x \ln \left (x \right )+18 c_{1} x -16 x^{2}+3 \sqrt {9 \ln \left (x \right )^{2} x^{2}+18 \ln \left (x \right ) c_{1} x^{2}+9 c_{1}^{2} x^{2}+18 x \ln \left (x \right )+18 c_{1} x -32 x^{2}+9}+9\right )\right )}^{\frac {1}{3}}}-\frac {2 x}{3 \left (x \ln \left (x \right )+c_{1} x +1\right )} \] Verified OK.
\[ y = -\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} \] Verified OK.
\[ y = -\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} \] Verified OK.
Writing the ode as \begin {align*} y^{\prime }&=\frac {y^{4} \left (x -1\right )}{x^{2} \left (2 y^{2}-1\right )}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is known. It is of type separable
. Therefore we do not need
to solve the PDE (A), and can just use the lookup table shown below to find \(\xi ,\eta \)
ODE class |
Form |
\(\xi \) |
\(\eta \) |
linear ode |
\(y'=f(x) y(x) +g(x)\) |
\(0\) |
\(e^{\int fdx}\) |
separable ode |
\(y^{\prime }=f\left ( x\right ) g\left ( y\right ) \) |
\(\frac {1}{f}\) |
\(0\) |
quadrature ode |
\(y^{\prime }=f\left ( x\right ) \) |
\(0\) |
\(1\) |
quadrature ode |
\(y^{\prime }=g\left ( y\right ) \) |
\(1\) |
\(0\) |
homogeneous ODEs of Class A |
\(y^{\prime }=f\left ( \frac {y}{x}\right ) \) |
\(x\) |
\(y\) |
homogeneous ODEs of Class C |
\(y^{\prime }=\left ( a+bx+cy\right ) ^{\frac {n}{m}}\) |
\(1\) |
\(-\frac {b}{c}\) |
homogeneous class D |
\(y^{\prime }=\frac {y}{x}+g\left ( x\right ) F\left (\frac {y}{x}\right ) \) |
\(x^{2}\) |
\(xy\) |
First order special form ID 1 |
\(y^{\prime }=g\left ( x\right ) e^{h\left (x\right ) +by}+f\left ( x\right ) \) |
\(\frac {e^{-\int bf\left ( x\right )dx-h\left ( x\right ) }}{g\left ( x\right ) }\) |
\(\frac {f\left ( x\right )e^{-\int bf\left ( x\right ) dx-h\left ( x\right ) }}{g\left ( x\right ) }\) |
polynomial type ode |
\(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\) |
\(\frac {a_{1}b_{2}x-a_{2}b_{1}x-b_{1}c_{2}+b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\) |
\(\frac {a_{1}b_{2}y-a_{2}b_{1}y-a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\) |
Bernoulli ode |
\(y^{\prime }=f\left ( x\right ) y+g\left ( x\right ) y^{n}\) |
\(0\) |
\(e^{-\int \left ( n-1\right ) f\left ( x\right ) dx}y^{n}\) |
Reduced Riccati |
\(y^{\prime }=f_{1}\left ( x\right ) y+f_{2}\left ( x\right )y^{2}\) |
\(0\) |
\(e^{-\int f_{1}dx}\) |
|
|||
|
|||
The above table shows that \begin {align*} \xi \left (x,y\right ) &=\frac {x^{2}}{x -1}\\ \tag {A1} \eta \left (x,y\right ) &=0 \end {align*}
The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Since \(\eta =0\) then in this special case \begin {align*} R = y \end {align*}
\(S\) is found from \begin {align*} S &= \int { \frac {1}{\xi }} dx\\ &= \int { \frac {1}{\frac {x^{2}}{x -1}}} dx \end {align*}
Which results in \begin {align*} S&= \frac {1}{x}+\ln \left (x \right ) \end {align*}
Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating \begin {align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end {align*}
Where in the above \(R_{x},R_{y},S_{x},S_{y}\) are all partial derivatives and \(\omega (x,y)\) is the right hand side of the original ode given by \begin {align*} \omega (x,y) &= \frac {y^{4} \left (x -1\right )}{x^{2} \left (2 y^{2}-1\right )} \end {align*}
Evaluating all the partial derivatives gives \begin {align*} R_{x} &= 0\\ R_{y} &= 1\\ S_{x} &= \frac {x -1}{x^{2}}\\ S_{y} &= 0 \end {align*}
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates. \begin {align*} \frac {dS}{dR} &= \frac {2 y^{2}-1}{y^{4}}\tag {2A} \end {align*}
We now need to express the RHS as function of \(R\) only. This is done by solving for \(x,y\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives \begin {align*} \frac {dS}{dR} &= \frac {2 R^{2}-1}{R^{4}} \end {align*}
The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates \(R,S\). Integrating the above gives \begin {align*} S \left (R \right ) = \frac {1}{3 R^{3}}-\frac {2}{R}+c_{1}\tag {4} \end {align*}
To complete the solution, we just need to transform (4) back to \(x,y\) coordinates. This results in \begin {align*} \frac {x \ln \left (x \right )+1}{x} = \frac {1}{3 y^{3}}-\frac {2}{y}+c_{1} \end {align*}
Which simplifies to \begin {align*} \frac {x \ln \left (x \right )+1}{x} = \frac {1}{3 y^{3}}-\frac {2}{y}+c_{1} \end {align*}
The following diagram shows solution curves of the original ode and how they transform in the canonical coordinates space using the mapping shown.
Original ode in \(x,y\) coordinates |
Canonical coordinates transformation |
ODE in canonical coordinates \((R,S)\) |
\( \frac {dy}{dx} = \frac {y^{4} \left (x -1\right )}{x^{2} \left (2 y^{2}-1\right )}\) |
|
\( \frac {d S}{d R} = \frac {2 R^{2}-1}{R^{4}}\) |
|
\(\!\begin {aligned} R&= y\\ S&= \frac {x \ln \left (x \right )+1}{x} \end {aligned} \) |
|
Summary
The solution(s) found are the following \begin{align*} \tag{1} \frac {x \ln \left (x \right )+1}{x} &= \frac {1}{3 y^{3}}-\frac {2}{y}+c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {x \ln \left (x \right )+1}{x} = \frac {1}{3 y^{3}}-\frac {2}{y}+c_{1} \] Verified OK.
Entering Exact first order ODE solver. (Form one type)
To solve an ode of the form\begin {equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A} \end {equation} We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \] Hence\begin {equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B} \end {equation} Comparing (A,B) shows that\begin {align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end {align*}
But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\] If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is \[ M(x,y) \mathop {\mathrm {d}x}+ N(x,y) \mathop {\mathrm {d}y}=0 \tag {1A} \] Therefore \begin {align*} \left (\frac {2 y^{2}-1}{y^{4}}\right )\mathop {\mathrm {d}y} &= \left (\frac {x -1}{x^{2}}\right )\mathop {\mathrm {d}x}\\ \left (-\frac {x -1}{x^{2}}\right )\mathop {\mathrm {d}x} + \left (\frac {2 y^{2}-1}{y^{4}}\right )\mathop {\mathrm {d}y} &= 0 \tag {2A} \end {align*}
Comparing (1A) and (2A) shows that \begin {align*} M(x,y) &= -\frac {x -1}{x^{2}}\\ N(x,y) &= \frac {2 y^{2}-1}{y^{4}} \end {align*}
The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied \[ \frac {\partial M}{\partial y} = \frac {\partial N}{\partial x} \] Using result found above gives \begin {align*} \frac {\partial M}{\partial y} &= \frac {\partial }{\partial y} \left (-\frac {x -1}{x^{2}}\right )\\ &= 0 \end {align*}
And \begin {align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (\frac {2 y^{2}-1}{y^{4}}\right )\\ &= 0 \end {align*}
Since \(\frac {\partial M}{\partial y}= \frac {\partial N}{\partial x}\), then the ODE is exact The following equations are now set up to solve for the function \(\phi \left (x,y\right )\) \begin {align*} \frac {\partial \phi }{\partial x } &= M\tag {1} \\ \frac {\partial \phi }{\partial y } &= N\tag {2} \end {align*}
Integrating (1) w.r.t. \(x\) gives \begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int M\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int -\frac {x -1}{x^{2}}\mathop {\mathrm {d}x} \\ \tag{3} \phi &= -\frac {1}{x}-\ln \left (x \right )+ f(y) \\ \end{align*} Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives \begin{equation} \tag{4} \frac {\partial \phi }{\partial y} = 0+f'(y) \end{equation} But equation (2) says that \(\frac {\partial \phi }{\partial y} = \frac {2 y^{2}-1}{y^{4}}\). Therefore equation (4) becomes \begin{equation} \tag{5} \frac {2 y^{2}-1}{y^{4}} = 0+f'(y) \end{equation} Solving equation (5) for \( f'(y)\) gives \[ f'(y) = \frac {2 y^{2}-1}{y^{4}} \] Integrating the above w.r.t \(y\) gives \begin{align*} \int f'(y) \mathop {\mathrm {d}y} &= \int \left ( \frac {2 y^{2}-1}{y^{4}}\right ) \mathop {\mathrm {d}y} \\ f(y) &= \frac {1}{3 y^{3}}-\frac {2}{y}+ c_{1} \\ \end{align*} Where \(c_{1}\) is constant of integration. Substituting result found above for \(f(y)\) into equation (3) gives \(\phi \) \[ \phi = -\frac {1}{x}-\ln \left (x \right )+\frac {1}{3 y^{3}}-\frac {2}{y}+ c_{1} \] But since \(\phi \) itself is a constant function, then let \(\phi =c_{2}\) where \(c_{2}\) is new constant and combining \(c_{1}\) and \(c_{2}\) constants into new constant \(c_{1}\) gives the solution as \[ c_{1} = -\frac {1}{x}-\ln \left (x \right )+\frac {1}{3 y^{3}}-\frac {2}{y} \]
The solution(s) found are the following \begin{align*} \tag{1} -\frac {1}{x}-\ln \left (x \right )+\frac {1}{3 y^{3}}-\frac {2}{y} &= c_{1} \\ \end{align*}
Verification of solutions
\[ -\frac {1}{x}-\ln \left (x \right )+\frac {1}{3 y^{3}}-\frac {2}{y} = c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {\left (x -1\right ) y^{5}}{x^{2} \left (-y+2 y^{3}\right )}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (x -1\right ) y^{5}}{x^{2} \left (-y+2 y^{3}\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \left (-y+2 y^{3}\right )}{y^{5}}=\frac {x -1}{x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \left (-y+2 y^{3}\right )}{y^{5}}d x =\int \frac {x -1}{x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {1}{3 y^{3}}-\frac {2}{y}=\frac {1}{x}+\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\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 \left (x \ln \left (x \right )+c_{1} x +1\right )}+\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 )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ 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*}