5.17 problem 4 and 16(iv)

Internal problem ID [12966]
Internal file name [OUTPUT/11619_Tuesday_November_07_2023_11_52_09_PM_98692280/index.tex]

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: 4 and 16(iv).
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {w^{\prime }-w \cos \left (w\right )=0} \] With initial conditions \begin {align*} [w \left (0\right ) = -1] \end {align*}

5.17.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} w^{\prime } &= f(t,w)\\ &= w \cos \left (w \right ) \end {align*}

The \(w\) domain of \(f(t,w)\) when \(t=0\) is \[ \{-\infty

The \(w\) domain of \(\frac {\partial f}{\partial w}\) when \(t=0\) is \[ \{-\infty

5.17.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int _{0}^{w}\frac {1}{\textit {\_a} \cos \left (\textit {\_a} \right )}d \textit {\_a} +\int _{0}^{-1}\frac {1}{w \cos \left (w \right )}d \textit {\_a} = t \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{0}^{w}\frac {1}{\textit {\_a} \cos \left (\textit {\_a} \right )}d \textit {\_a} +\int _{0}^{-1}\frac {1}{w \cos \left (w \right )}d \textit {\_a} &= t \\ \end{align*}

Verification of solutions

\[ \int _{0}^{w}\frac {1}{\textit {\_a} \cos \left (\textit {\_a} \right )}d \textit {\_a} +\int _{0}^{-1}\frac {1}{w \cos \left (w \right )}d \textit {\_a} = t \] Verified OK.

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.109 (sec). Leaf size: 19

dsolve([diff(w(t),t)=w(t)*cos( w(t)),w(0) = -1],w(t), singsol=all)
 

\[ w \left (t \right ) = \operatorname {RootOf}\left (\int _{\textit {\_Z}}^{-1}\frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} +t \right ) \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[{w'[t]==w[t]*Cos[ w[t]],{w[0]==-1}},w[t],t,IncludeSingularSolutions -> True]
 

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