2.17 problem 17

Internal problem ID [5012]

Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number: 17.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]

Solve \begin {gather*} \boxed {\frac {1}{x^{2}-x y+y^{2}}-\frac {y^{\prime }}{2 y^{2}-x y}=0} \end {gather*}

✓ Solution by Maple

Time used: 0.686 (sec). Leaf size: 40

dsolve(1/(x^2-x*y(x)+y(x)^2)=1/(2*y(x)^2-x*y(x))*diff(y(x),x),y(x), singsol=all)
 

\[ y \relax (x ) = \left (\RootOf \left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}-\textit {\_Z}^{4}-2 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x \]

✓ Solution by Mathematica

Time used: 0.095 (sec). Leaf size: 1805

DSolve[1/(x^2-x*y[x]+y[x]^2)==1/(2*y[x]^2-x*y[x])*y'[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}-\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2+\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}+\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2+\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}-\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2-\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}+\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2-\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ \end{align*}