Internal problem ID [3448]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 710.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {\left (4 x -x y^{3}-2 y^{4}\right ) y^{\prime }-\left (2+y^{3}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 27
dsolve((4*x-x*y(x)^3-2*y(x)^4)*diff(y(x),x) = (2+y(x)^3)*y(x),y(x), singsol=all)
\[ x -\frac {\left (-y \relax (x )^{2}+c_{1}\right ) y \relax (x )^{2}}{2+y \relax (x )^{3}} = 0 \]
✓ Solution by Mathematica
Time used: 165.976 (sec). Leaf size: 2062
DSolve[(4 x-x y[x]^3-2 y[x]^4)y'[x]==(2+y[x]^3)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}+\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}+\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}-\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}-\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to 0 \\ y(x)\to \sqrt [3]{-2} \\ y(x)\to -\sqrt [3]{2} \\ y(x)\to -(-1)^{2/3} \sqrt [3]{2} \\ \end{align*}