72.5.28 problem 24

Internal problem ID [14635]
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 1.6 page 89
Problem number : 24
Date solved : Thursday, March 13, 2025 at 04:11:19 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y^{2}-4 y+2 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-2 \end{align*}

✓ Maple. Time used: 0.145 (sec). Leaf size: 24

ode:=diff(y(t),t) = y(t)^2-4*y(t)+2; 
ic:=y(0) = -2; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = 2-\sqrt {2}\, \tanh \left (\operatorname {arctanh}\left (2 \sqrt {2}\right )+\sqrt {2}\, t \right ) \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 59

ode=D[y[t],t]==y[t]^2-4*y[t]+2; 
ic={y[0]==-2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {2 \left (\left (\sqrt {2}-3\right ) e^{2 \sqrt {2} t}+3+\sqrt {2}\right )}{\left (4+\sqrt {2}\right ) e^{2 \sqrt {2} t}-4+\sqrt {2}} \]

✓ Sympy. Time used: 0.459 (sec). Leaf size: 68

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

\[ t - \frac {\sqrt {2} \log {\left (y{\left (t \right )} - 2 - \sqrt {2} \right )}}{4} + \frac {\sqrt {2} \log {\left (y{\left (t \right )} - 2 + \sqrt {2} \right )}}{4} = - \frac {\sqrt {2} \log {\left (\sqrt {2} + 4 \right )}}{4} + \frac {\sqrt {2} \log {\left (4 - \sqrt {2} \right )}}{4} \]