63.5.30 problem 15(e)

Internal problem ID [13104]
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)
Date solved : Tuesday, January 28, 2025 at 04:52:31 AM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=a x+b x^{3} \end{align*}

✓ Solution by Maple

Time used: 0.008 (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: 0.217 (sec). Leaf size: 75

DSolve[D[x[t],t]==a*x[t]+b*x[t]^3,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (b K[1]^2+a\right )}dK[1]\&\right ][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*}