Internal problem ID [10410]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY.
2015.
Section: Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page
41
Problem number: 15(e).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_quadrature]
\[ \boxed {x^{\prime }-a x-b x^{3}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 70
dsolve(diff(x(t),t)=a*x(t)+b*x(t)^3,x(t), singsol=all)
\begin{align*} x \left (t \right ) = \frac {\sqrt {\left (c_{1} a \,{\mathrm e}^{-2 a t}-b \right ) a}}{c_{1} a \,{\mathrm e}^{-2 a t}-b} \\ x \left (t \right ) = -\frac {\sqrt {\left (c_{1} a \,{\mathrm e}^{-2 a t}-b \right ) a}}{c_{1} a \,{\mathrm e}^{-2 a t}-b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.781 (sec). Leaf size: 118
DSolve[x'[t]==a*x[t]+b*x[t]^3,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {i \sqrt {a} e^{a (t+c_1)}}{\sqrt {-1+b e^{2 a (t+c_1)}}} \\ x(t)\to \frac {i \sqrt {a} e^{a (t+c_1)}}{\sqrt {-1+b e^{2 a (t+c_1)}}} \\ x(t)\to 0 \\ x(t)\to -\frac {i \sqrt {a}}{\sqrt {b}} \\ x(t)\to \frac {i \sqrt {a}}{\sqrt {b}} \\ \end{align*}