68.4.36 problem 36

Internal problem ID [17216]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 36
Date solved : Thursday, October 02, 2025 at 01:56:57 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y^{3}-1 \end{align*}

✓ Maple. Time used: 0.186 (sec). Leaf size: 53

ode:=diff(y(t),t) = y(t)^3-1; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (-\sqrt {3}\, \ln \left (\cos \left (\textit {\_Z} \right )^{2}\right )-2 \sqrt {3}\, \ln \left (\tan \left (\textit {\_Z} \right )-\sqrt {3}\right )+6 \sqrt {3}\, c_1 +6 \sqrt {3}\, t +6 \textit {\_Z} \right )\right )}{2}-\frac {1}{2} \]

✓ Mathematica. Time used: 0.128 (sec). Leaf size: 61

ode=D[y[t],t]==y[t]^3-1; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) \left (K[1]^2+K[1]+1\right )}dK[1]\&\right ][t+c_1]\\ y(t)&\to 1\\ y(t)&\to -\sqrt [3]{-1}\\ y(t)&\to (-1)^{2/3} \end{align*}

✓ Sympy. Time used: 0.367 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t)**3 + Derivative(y(t), t) + 1,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ t - \frac {\log {\left (y{\left (t \right )} - 1 \right )}}{3} + \frac {\log {\left (y^{2}{\left (t \right )} + y{\left (t \right )} + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (2 y{\left (t \right )} + 1\right )}{3} \right )}}{3} = C_{1} \]