Internal problem ID [4564]
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]
\[ \boxed {\left (2 y^{3} x -x^{4}\right ) y^{\prime }+2 y x^{3}-y^{4}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 317
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 \left (x \right ) &= \frac {12^{\frac {1}{3}} \left (x 12^{\frac {1}{3}} c_{1} +{\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {2}{3}}\right )}{6 c_{1} {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) c_{1} 2^{\frac {2}{3}} x \right ) 2^{\frac {2}{3}}}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} \left (\left (1-i \sqrt {3}\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) c_{1} 2^{\frac {2}{3}} x \right ) 2^{\frac {2}{3}}}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.224 (sec). Leaf size: 331
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 {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 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]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\ \end{align*}