Internal problem ID [8635]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 299.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _exact, _rational]
\[ \boxed {\left (3 x y^{2}-x^{2}\right ) y^{\prime }+y^{3}-2 y x=0} \]
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 242
dsolve((3*x*y(x)^2-x^2)*diff(y(x),x)+y(x)^3-2*x*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {12^{\frac {1}{3}} \left (x^{3} 12^{\frac {1}{3}}+{\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{\frac {2}{3}}\right )}{6 x {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) x^{3} 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (\left (1-i \sqrt {3}\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{\frac {2}{3}}+x^{3} 2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )\right )}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 34.309 (sec). Leaf size: 328
DSolve[(3*x*y[x]^2-x^2)*y'[x]+y[x]^3-2*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ \end{align*}