Internal problem ID [3409]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 670.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (1-x^{3}+6 x^{2} y^{2}\right ) y^{\prime }-\left (6+3 y x -4 y^{3}\right ) x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 601
dsolve((1-x^3+6*x^2*y(x)^2)*diff(y(x),x) = (6+3*x*y(x)-4*y(x)^3)*x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}{6 x}+\frac {x^{3}-1}{x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}{12 x}-\frac {x^{3}-1}{2 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}{6 x}-\frac {x^{3}-1}{x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}{12 x}-\frac {x^{3}-1}{2 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}{6 x}-\frac {x^{3}-1}{x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.734 (sec). Leaf size: 424
DSolve[(1-x^3+6 x^2 y[x]^2)y'[x]==(6+3 x y[x]-4 y[x]^3)x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{2} \left (x^3-1\right )}{\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}-\frac {\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{6 \sqrt [3]{2} x^2} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2} \\ \end{align*}