72.5.25 problem 12

Internal problem ID [14711]
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 : Tuesday, January 28, 2025 at 07:12:19 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[D[w[t],t]==(w[t]^2-2)*Arctan[ w[t]],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*}