Internal problem ID [3949]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 703.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {x \left (a +b x y^{3}\right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 492
dsolve(x*(a+b*x*y(x)^3)*diff(y(x),x)+(a+c*x^3*y(x))*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (-x^{2} b \left (c \,x^{2}-2 c_{1} \right ) 3^{\frac {1}{3}}+{\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {2}{3}}\right )}{3 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {1}{3}} b x} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {2}{3}}+x^{2} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) b \left (c \,x^{2}-2 c_{1} \right )\right ) 3^{\frac {1}{3}}}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {1}{3}} b x} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {2}{3}}+x^{2} b \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) \left (c \,x^{2}-2 c_{1} \right )\right )}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {1}{3}} b x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.307 (sec). Leaf size: 484
DSolve[x(a+b x y[x]^3)y'[x]+(a+c x^3 y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (-c x^2+2 c_1\right )}{\sqrt [3]{3} \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}+\frac {\sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}{3^{2/3} b x} \\ y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}+\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) b x^2 \left (c x^2-2 c_1\right )}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}} \\ y(x)\to \frac {\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) b x^2 \left (c x^2-2 c_1\right )-i \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}} \\ \end{align*}