Internal problem ID [4056]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous
Methods
Problem number: Exercise 12.43, page 103.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 447
dsolve((2*x*y(x)^3-x^4)*diff(y(x),x)+2*x^3*y(x)-y(x)^4=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {12^{\frac {1}{3}} \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}+\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {12^{\frac {1}{3}} \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {x 12^{\frac {2}{3}}}{12 \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {12^{\frac {1}{3}} \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {12^{\frac {1}{3}} \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {x 12^{\frac {2}{3}}}{12 \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {12^{\frac {1}{3}} \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 46.074 (sec). Leaf size: 383
DSolve[(2*x*y[x]^3-x^4)*y'[x]+2*x^3*y[x]-y[x]^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \sqrt [3]{-3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\ y(x)\to \frac {-\sqrt [3]{-2} \sqrt [6]{3} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-\left (\left (\sqrt {3}-3 i\right ) e^{c_1} x\right )}{2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\ y(x)\to \frac {\sqrt [3]{\sqrt {x^6}-x^3}}{\sqrt [3]{2}} \\ y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{\sqrt {x^6}-x^3} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\sqrt {x^6}-x^3}}{\sqrt [3]{2}} \\ \end{align*}