5.14 problem 4 and 16(i)

Internal problem ID [12963]
Internal file name [OUTPUT/11616_Tuesday_November_07_2023_11_52_06_PM_51224347/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(i).
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 ) = 0] \end {align*}

5.14.1 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

5.14.2 Solving as quadrature ode

Since ode has form \(w^{\prime }= f(w)\) and initial conditions \(w = 0\) is verified to satisfy the ode, then the solution is \begin {align*} w&=w_0 \\ &=0 \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} w &= 0 \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ w = 0 \] Verified OK.

5.14.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [w^{\prime }-w \cos \left (w\right )=0, w \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & w^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & w^{\prime }=w \cos \left (w\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {w^{\prime }}{w \cos \left (w\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {w^{\prime }}{w \cos \left (w\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {w^{\prime }}{w \cos \left (w\right )}d t =t +c_{1} \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \end {array} \]

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: 0.0 (sec). Leaf size: 5

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

\[ w \left (t \right ) = 0 \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 6

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

\[ w(t)\to 0 \]