Internal problem ID [7823]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 243.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {x \left (2 y+x -1\right ) y^{\prime }-y \left (y+2 x +1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 493
dsolve(x*(2*y(x)+x-1)*diff(y(x),x)-y(x)*(y(x)+2*x+1)=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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{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 -x +80 c_{1}}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 38.182 (sec). Leaf size: 463
DSolve[x*(2*y[x]+x-1)*y'[x]-y[x]*(y[x]+2*x+1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{2} x}{\sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}+\frac {\sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{3 \sqrt [3]{2} c_1}+x-1 \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}+x-1 \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}+x-1 \\ y(x)\to \text {Indeterminate} \\ y(x)\to x-1 \\ \end{align*}