Internal problem ID [2164]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page
79
Problem number: Problem 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{2}+2 y x -2 x^{2}}{x^{2}-y x +y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.39 (sec). Leaf size: 79
dsolve(diff(y(x),x)=(y(x)^2+2*x*y(x)-2*x^2)/(x^2-x*y(x)+y(x)^2),y(x), singsol=all)
\[ y \relax (x ) = -\frac {x \left (\RootOf \left (2 \textit {\_Z}^{6}+\left (9 x^{2} c_{1}-1\right ) \textit {\_Z}^{4}-6 x^{2} c_{1} \textit {\_Z}^{2}+x^{2} c_{1}\right )^{2}-1\right )}{\RootOf \left (2 \textit {\_Z}^{6}+\left (9 x^{2} c_{1}-1\right ) \textit {\_Z}^{4}-6 x^{2} c_{1} \textit {\_Z}^{2}+x^{2} c_{1}\right )^{2}} \]
✓ Solution by Mathematica
Time used: 60.187 (sec). Leaf size: 372
DSolve[y'[x]==(y[x]^2+2*x*y[x]-2*x^2)/(x^2-x*y[x]+y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (-3 x^2+e^{2 c_1}\right )}{\sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (-3 x^2+e^{2 c_1}\right )}{2^{2/3} \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+\left (-\frac {1}{3}\right )^{2/3} \sqrt [3]{-9 x^3+\sqrt {3} \sqrt {27 e^{2 c_1} x^4-9 e^{4 c_1} x^2+e^{6 c_1}}}+x \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-3 x^2+e^{2 c_1}\right )}{2^{2/3} \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ \end{align*}