Internal problem ID [8659]
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]
\[ \boxed {\left (a x y^{3}+c \right ) x y^{\prime }+\left (b \,x^{3} y+c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 492
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 \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (-a \,x^{2} \left (b \,x^{2}-2 c_{1} \right ) 3^{\frac {1}{3}}+{\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{\frac {2}{3}}\right )}{3 {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{\frac {1}{3}} a x} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{\frac {2}{3}}+a \,x^{2} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (b \,x^{2}-2 c_{1} \right )\right ) 3^{\frac {1}{3}}}{6 {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{\frac {1}{3}} a x} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) a \,x^{2} \left (b \,x^{2}-2 c_{1} \right )\right ) 3^{\frac {1}{3}}}{6 {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{\frac {1}{3}} a x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 54.413 (sec). Leaf size: 484
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 {i \sqrt [3]{3} \left (\sqrt {3}+i\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}+\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) 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 )}}} \\ y(x)\to \frac {\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) a x^2 \left (b x^2-2 c_1\right )-i \sqrt [3]{3} \left (\sqrt {3}-i\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}}{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*}