29.25.13 problem 710

Internal problem ID [5300]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 710
Date solved : Monday, January 27, 2025 at 11:05:07 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (4 x -x y^{3}-2 y^{4}\right ) y^{\prime }&=\left (2+y^{3}\right ) y \end{align*}

✓ Solution by Maple

Time used: 0.013 (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)
 

\[ x -\frac {\left (-y \left (x \right )^{2}+c_{1} \right ) y \left (x \right )^{2}}{2+y \left (x \right )^{3}} = 0 \]

✓ Solution by Mathematica

Time used: 60.215 (sec). Leaf size: 2021

DSolve[(4 x-x y[x]^3-2 y[x]^4)D[y[x],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} \\ \end{align*}