Internal problem ID [10780]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {y^{\prime }-\frac {y}{x +y^{3}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 260
dsolve(diff(y(x),x)=y(x)/(x+y(x)^3),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}-\frac {2 c_{1}}{\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) = -\frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {c_{1}}{\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}+\frac {2 c_{1}}{\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \left (x \right ) = -\frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {c_{1}}{\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}+\frac {2 c_{1}}{\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.733 (sec). Leaf size: 227
DSolve[y'[x]==y[x]/(x+y[x]^3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2\ 3^{2/3} c_1-\sqrt [3]{3} \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}}{3 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to \frac {-(-1)^{2/3} \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [3]{-3} c_1}{3^{2/3} \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to \frac {2 \sqrt [3]{-3} \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}+4 (-3)^{2/3} c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to 0 \\ \end{align*}