Internal problem ID [3442]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 704.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {x \left (1-2 x^{2} y^{3}\right ) y^{\prime }+\left (1-2 x^{3} y^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 770
dsolve(x*(1-2*x^2*y(x)^3)*diff(y(x),x)+(1-2*x^3*y(x)^2)*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \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 (c_{1}-2 x \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 {c_{1}}{6}-\frac {x}{3} \\ y \relax (x ) = -\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 (c_{1}-2 x \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 {c_{1}}{6}-\frac {x}{3}-\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 (c_{1}-2 x \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 \relax (x ) = -\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 (c_{1}-2 x \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 {c_{1}}{6}-\frac {x}{3}+\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 (c_{1}-2 x \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: 9.237 (sec). Leaf size: 582
DSolve[x(1-2 x^2 y[x]^3)y'[x]+(1-2 x^3 y[x]^2)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 (27+2 x^2 (2 x-c_1){}^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 (27+2 x^2 (2 x-c_1){}^3\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 (27+2 x^2 (2 x-c_1){}^3\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 (27+2 x^2 (2 x-c_1){}^3\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 (27+2 x^2 (2 x-c_1){}^3\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 (27+2 x^2 (2 x-c_1){}^3\right )}}}{12 x^2} \\ \end{align*}