72.5.25 problem 12

Internal problem ID [14632]
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 : 12
Date solved : Thursday, March 13, 2025 at 04:11:09 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 25

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

\[ t -\int _{}^{w}\frac {1}{\left (\textit {\_a}^{2}-2\right ) \arctan \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} = 0 \]

✓ Mathematica. Time used: 0.563 (sec). Leaf size: 62

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

\begin{align*} w(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\text {Arctan}(K[1]) \left (K[1]^2-2\right )}dK[1]\&\right ][t+c_1] \\ w(t)\to -\sqrt {2} \\ w(t)\to \sqrt {2} \\ w(t)\to \text {Arctan}^{(-1)}(0) \\ \end{align*}

✓ Sympy. Time used: 1.055 (sec). Leaf size: 17

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

\[ - \int \limits ^{w{\left (t \right )}} \frac {1}{\left (y^{2} - 2\right ) \operatorname {atan}{\left (y \right )}}\, dy = C_{1} - t \]