Internal problem ID [15148]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 12. Miscellaneous problems. Exercises page 93
Problem number: 286.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {\frac {1}{x^{2}-y x +y^{2}}-\frac {y^{\prime }}{2 y^{2}-y x}=0} \]
✓ Solution by Maple
Time used: 0.938 (sec). Leaf size: 40
dsolve(1/(x^2-x*y(x)+y(x)^2)=diff(y(x),x)/(2*y(x)^2-x*y(x)),y(x), singsol=all)
\[ y = \left (\operatorname {RootOf}\left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}-\textit {\_Z}^{4}-2 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x \]
✓ Solution by Mathematica
Time used: 60.201 (sec). Leaf size: 1805
DSolve[1/(x^2-x*y[x]+y[x]^2)==y'[x]/(2*y[x]^2-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*}