1.9 problem 15

Internal problem ID [13733]

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: 0.125 (sec). Leaf size: 2300

dsolve((2*x-y(x))-y(x)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) = -\frac {8 x \left (\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}-\frac {4 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}-1\right )}{4 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}} y \left (x \right ) = -\frac {8 x \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (4 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )}{4}-1\right )}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (4 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )} y \left (x \right ) = -\frac {8 x \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (4 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )}{4}-1\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (4 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )} y \left (x \right ) = -\frac {8 x \left (-\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}-i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )-1\right )}{-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )} y \left (x \right ) = -\frac {8 x \left (-\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}+i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )-1\right )}{-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )} y \left (x \right ) = -\frac {8 x \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )}{4}-1\right )}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )} y \left (x \right ) = -\frac {8 x \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )}{4}-1\right )}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )} y \left (x \right ) = -\frac {8 x \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )}{4}-1\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )} y \left (x \right ) = -\frac {8 x \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )}{4}-1\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (-2 \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}\right )\right )} \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*}