Internal problem ID [3903]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 656.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {3 x \left (x +y^{2}\right ) y^{\prime }-3 y x -2 y^{3}=-x^{3}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 450
dsolve(3*x*(x+y(x)^2)*diff(y(x),x)+x^3-3*x*y(x)-2*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}} y \left (x \right ) = -\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 28.201 (sec). Leaf size: 362
DSolve[3 x(x+y[x]^2)y'[x]+x^3-3 x y[x]-2 y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}} y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 \sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}} y(x)\to \frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x-i 2^{2/3} \left (\sqrt {3}-i\right ) \left (-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}\right ){}^{2/3}}{4 \sqrt [3]{-x^3+c_1 x^2+\sqrt {x^3 \left (x^3-2 c_1 x^2+c_1{}^2 x+4\right )}}} \end{align*}