Internal problem ID [12128]
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: 0.016 (sec). Leaf size: 224
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 {2}{3}}-6 c_{1}}{3 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 i \sqrt {3}\, c_{1} +\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}-6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 i \sqrt {3}\, c_{1} -\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.895 (sec). Leaf size: 263
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 {\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to \frac {\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to 0 \\ \end{align*}