72.8.4 problem 5

Internal problem ID [14697]
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. Review Exercises for chapter 1. page 136
Problem number : 5
Date solved : Monday, March 31, 2025 at 12:54:03 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {\left (t^{2}-4\right ) \left (1+y\right ) {\mathrm e}^{y}}{\left (t -1\right ) \left (3-y\right )} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 38

ode:=diff(y(t),t) = (t^2-4)*(y(t)+1)*exp(y(t))/(t-1)/(3-y(t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\operatorname {RootOf}\left (8 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (1-\textit {\_Z} \right )+t^{2}-6 \ln \left (t -1\right )-2 \,{\mathrm e}^{\textit {\_Z}}+2 c_1 +2 t \right ) \]

✓ Mathematica. Time used: 1.164 (sec). Leaf size: 62

ode=D[y[t],t]==(  (t^2-4)*(1+y[t])*Exp[y[t]])/(   (t-1)*(3-y[t])); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{-K[1]} (K[1]-3)}{K[1]+1}dK[1]\&\right ]\left [\int _1^t\frac {4-K[2]^2}{K[2]-1}dK[2]+c_1\right ] \\ y(t)\to -1 \\ \end{align*}

✓ Sympy. Time used: 1.636 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), t) - (t**2 - 4)*(y(t) + 1)*exp(y(t))/((3 - y(t))*(t - 1)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \int \limits ^{y{\left (t \right )}} \frac {\left (y - 3\right ) e^{- y}}{y + 1}\, dy = C_{1} - \frac {t^{2}}{2} - t + 3 \log {\left (t - 1 \right )} \]