Internal problem ID [8660]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 324.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {\left (2 y^{3} x^{3}-x \right ) y^{\prime }+2 y^{3} x^{3}-y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 770
dsolve((2*x^3*y(x)^3-x)*diff(y(x),x)+2*x^3*y(x)^3-y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}{6 x}+\frac {\left (-2 x +c_{1} \right )^{2} x}{6 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}-\frac {x}{3}+\frac {c_{1}}{6} y \left (x \right ) = -\frac {{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}{12 x}-\frac {\left (-2 x +c_{1} \right )^{2} x}{12 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}-\frac {x}{3}+\frac {c_{1}}{6}-\frac {i \sqrt {3}\, \left (\frac {{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}{6 x}-\frac {\left (-2 x +c_{1} \right )^{2} x}{6 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}{12 x}-\frac {\left (-2 x +c_{1} \right )^{2} x}{12 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}-\frac {x}{3}+\frac {c_{1}}{6}+\frac {i \sqrt {3}\, \left (\frac {{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}{6 x}-\frac {\left (-2 x +c_{1} \right )^{2} x}{6 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 60.125 (sec). Leaf size: 672
DSolve[-y[x] + 2*x^3*y[x]^3 + (-x + 2*x^3*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 x^3+c_1 x^2+\frac {x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}+\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{6 x^2} y(x)\to \frac {2 x^2 (-2 x+c_1)-\frac {i \left (\sqrt {3}-i\right ) x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{12 x^2} y(x)\to \frac {2 x^2 (-2 x+c_1)+\frac {i \left (\sqrt {3}+i\right ) x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{12 x^2} \end{align*}