Internal problem ID [3441]
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]
Solve \begin {gather*} \boxed {x \left (a +b x y^{3}\right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 761
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 \relax (x ) = \frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{3 b x}+\frac {\left (-c \,x^{2}+2 c_{1}\right ) x}{\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{6 b x}-\frac {\left (-c \,x^{2}+2 c_{1}\right ) x}{2 \left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{3 b x}-\frac {\left (-c \,x^{2}+2 c_{1}\right ) x}{\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{6 b x}-\frac {\left (-c \,x^{2}+2 c_{1}\right ) x}{2 \left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{3 b x}-\frac {\left (-c \,x^{2}+2 c_{1}\right ) x}{\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 c^{2} c_{1} x^{6}-12 c c_{1}^{2} x^{4}+8 c_{1}^{3} x^{2}-27 b \,a^{2}\right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.232 (sec). Leaf size: 463
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 {\sqrt [3]{-\frac {1}{3}} x \left (c x^2-2 c_1\right )}{\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 {i \left (\sqrt {3}+i\right ) \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 )}}}{2\ 3^{2/3} b x} \\ y(x)\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\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}-2 (-3)^{2/3} 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 )}}} \\ \end{align*}