78.6.14 problem 5

Internal problem ID [21077]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 7, Nonlinear systems. Problems section 7.11
Problem number : 5
Date solved : Thursday, October 02, 2025 at 07:04:41 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=x^{3}+a x^{2}-b x \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 62

ode:=diff(x(t),t) = x(t)^3+a*x(t)^2-b*x(t); 
dsolve(ode,x(t), singsol=all);
 

\[ t -\frac {\ln \left (a x+x^{2}-b \right )}{2 b}+\frac {a \,\operatorname {arctanh}\left (\frac {2 x+a}{\sqrt {a^{2}+4 b}}\right )}{b \sqrt {a^{2}+4 b}}+\frac {\ln \left (x\right )}{b}+c_1 = 0 \]

✓ Mathematica. Time used: 0.398 (sec). Leaf size: 128

ode=D[x[t],t]==x[t]^3+a*x[t]^2-b*x[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \text {InverseFunction}\left [-\frac {\frac {2 a \arctan \left (\frac {2 \text {$\#$1}+a}{\sqrt {-a^2-4 b}}\right )}{\sqrt {-a^2-4 b}}+\log (\text {$\#$1} (\text {$\#$1}+a)-b)-2 \log (\text {$\#$1})}{2 b}\&\right ][-t+c_1]\\ x(t)&\to 0\\ x(t)&\to \frac {1}{2} \left (-\sqrt {a^2+4 b}-a\right )\\ x(t)&\to \frac {1}{2} \left (\sqrt {a^2+4 b}-a\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
x = Function("x") 
ode = Eq(a*x(t)**2 - b*x(t) - x(t)**3 - x(t) + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out