Internal problem ID [3981]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number: Recognizable Exact Differential equations. Integrating factors. Exercise
10.14, page 90.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y \left (2 x -y-1\right )+x \left (2 y-x -1\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 499
dsolve((y(x)*(2*x-y(x)-1))+(x*(2*y(x)-x-1))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{40 c_{1}}+\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}-x -1 \\ y \relax (x ) = -\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{80 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{80 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}-x -1-\frac {i \sqrt {3}\, \left (\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{40 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{80 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{80 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}-x -1+\frac {i \sqrt {3}\, \left (\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{40 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1}-x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.649 (sec). Leaf size: 448
DSolve[(y[x]*(2*x-y[x]-1))+(x*(2*y[x]-x-1))*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}-\frac {\sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}{\sqrt [3]{2} 3^{2/3} c_1}-x-1 \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}{2 \sqrt [3]{2} 3^{2/3} c_1}+\frac {x+i \sqrt {3} x}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}-x-1 \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}{2 \sqrt [3]{2} 3^{2/3} c_1}+\frac {x-i \sqrt {3} x}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}-x-1 \\ y(x)\to \text {Indeterminate} \\ y(x)\to -x-1 \\ \end{align*}