72.3.9 problem 9

Internal problem ID [14602]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.4 page 61
Problem number : 9
Date solved : Monday, March 31, 2025 at 12:39:55 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y^{2}-y^{3} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{5}} \end{align*}

✓ Maple. Time used: 0.567 (sec). Leaf size: 17

ode:=diff(y(t),t) = y(t)^2-y(t)^3; 
ic:=y(0) = 1/5; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {1}{\operatorname {LambertW}\left (4 \,{\mathrm e}^{4-t}\right )+1} \]

✓ Mathematica. Time used: 0.311 (sec). Leaf size: 31

ode=D[y[t],t]==y[t]^2-y[t]^3; 
ic={y[0]==2/10}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \text {InverseFunction}\left [\frac {1}{\text {$\#$1}}+\log (1-\text {$\#$1})-\log (\text {$\#$1})\&\right ][-t+5+\log (4)] \]

✓ Sympy. Time used: 0.309 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t)**3 - y(t)**2 + Derivative(y(t), t),0) 
ics = {y(0): 1/5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ t + \log {\left (y{\left (t \right )} - 1 \right )} - \log {\left (y{\left (t \right )} \right )} + \frac {1}{y{\left (t \right )}} = 2 \log {\left (2 \right )} + 5 + i \pi \]