Internal problem ID [14053]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {-y-y y^{\prime }=-2 x} \]
✓ Solution by Maple
Time used: 0.891 (sec). Leaf size: 1041
dsolve((2*x-y(x))-y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= x \left (-1+\frac {\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}}{2 c_{1} x}+\frac {2 c_{1} x}{\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}}\right ) \\ y \left (x \right ) &= \frac {4 c_{1}^{2} x^{2}-2 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}+\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{2 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 c_{1}^{2} x^{2}-2 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}+\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{2 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}} \sqrt {3}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}-4 c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}\right ) x}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}} \sqrt {3}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}-4 c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}\right ) x}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}} \sqrt {3}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}-4 c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}\right ) x}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 54.579 (sec). Leaf size: 496
DSolve[(2*x-y[x])-y[x]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}-x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}-x \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{2 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}-x \\ y(x)\to \sqrt [3]{x^3}+\frac {\left (x^3\right )^{2/3}}{x}-x \\ y(x)\to \frac {1}{2} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3}+\frac {\left (-1-i \sqrt {3}\right ) \left (x^3\right )^{2/3}}{x}-2 x\right ) \\ y(x)\to \frac {1}{2} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^3}+\frac {i \left (\sqrt {3}+i\right ) \left (x^3\right )^{2/3}}{x}-2 x\right ) \\ \end{align*}