Internal problem ID [7903]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 323.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\left (a x y^{3}+c \right ) x y^{\prime }+\left (b \,x^{3} y+c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 761
dsolve((a*x*y(x)^3+c)*x*diff(y(x),x)+(b*x^3*y(x)+c)*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}{3 a x}+\frac {\left (-b \,x^{2}+2 c_{1}\right ) x}{\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}{6 a x}-\frac {\left (-b \,x^{2}+2 c_{1}\right ) x}{2 \left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}{3 a x}-\frac {\left (-b \,x^{2}+2 c_{1}\right ) x}{\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}{6 a x}-\frac {\left (-b \,x^{2}+2 c_{1}\right ) x}{2 \left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}{3 a x}-\frac {\left (-b \,x^{2}+2 c_{1}\right ) x}{\left (\left (27 c +3 \sqrt {-\frac {3 \left (-b^{3} x^{8}+6 b^{2} x^{6} c_{1}-12 b \,x^{4} c_{1}^{2}+8 x^{2} c_{1}^{3}-27 a \,c^{2}\right )}{a}}\right ) a^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.36 (sec). Leaf size: 463
DSolve[y[x]*(c + b*x^3*y[x]) + x*(c + a*x*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (-b x^2+2 c_1\right )}{\sqrt [3]{3} \sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}}+\frac {\sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}}{3^{2/3} a x} \\ y(x)\to \frac {\sqrt [3]{-\frac {1}{3}} x \left (b x^2-2 c_1\right )}{\sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}}{2\ 3^{2/3} a x} \\ y(x)\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}-2 (-3)^{2/3} a x^2 \left (b x^2-2 c_1\right )}{6 a x \sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}} \\ \end{align*}