10.4.11 problem 13

Internal problem ID [1192]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.5. Page 88
Problem number : 13
Date solved : Saturday, March 29, 2025 at 10:45:28 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\left (1-y\right )^{2} y^{2} \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 66

ode:=diff(y(t),t) = (1-y(t))^2*y(t)^2; 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-2 \ln \left ({\mathrm e}^{\textit {\_Z}}+1\right ) {\mathrm e}^{2 \textit {\_Z}}+c_1 \,{\mathrm e}^{2 \textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{2 \textit {\_Z}}+t \,{\mathrm e}^{2 \textit {\_Z}}-2 \ln \left ({\mathrm e}^{\textit {\_Z}}+1\right ) {\mathrm e}^{\textit {\_Z}}+c_1 \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+t \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}+1\right )}+1 \]

✓ Mathematica. Time used: 0.339 (sec). Leaf size: 50

ode=D[y[t],t] == (1-y[t])^2*y[t]^2; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to \text {InverseFunction}\left [-\frac {1}{\text {$\#$1}-1}-\frac {1}{\text {$\#$1}}-2 \log (1-\text {$\#$1})+2 \log (\text {$\#$1})\&\right ][t+c_1] \\ y(t)\to 0 \\ y(t)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.875 (sec). Leaf size: 32

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

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