4.43 problem 43

Internal problem ID [14200]
Internal file name [OUTPUT/13881_Saturday_March_09_2024_03_56_36_PM_90253575/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number: 43.
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]

\[ \boxed {-\cos \left (y\right ) y^{\prime }=-1} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}

4.43.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= \frac {1}{\cos \left (y \right )} \end {align*}

The \(y\) domain of \(f(x,y)\) when \(x=0\) is \[ \left \{y <\frac {1}{2} \pi +\pi \_Z221 \boldsymbol {\lor }\frac {1}{2} \pi +\pi \_Z221

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(x=0\) is \[ \left \{y <\frac {1}{2} \pi +\pi \_Z221 \boldsymbol {\lor }\frac {1}{2} \pi +\pi \_Z221

4.43.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \cos \left (y \right )d y &= \int {dx}\\ \sin \left (y \right )&= x +c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=2\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} \sin \left (2\right ) = c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = \sin \left (2\right ) \end {align*}

Trying the constant \begin {align*} c_{1} = \sin \left (2\right ) \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} \sin \left (y \right ) = x +\sin \left (2\right ) \end {align*}

The constant \(c_{1} = \sin \left (2\right )\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} \sin \left (y\right ) &= x +\sin \left (2\right ) \\ \end{align*}

Verification of solutions

\[ \sin \left (y\right ) = x +\sin \left (2\right ) \] Verified OK.

4.43.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [-\cos \left (y\right ) y^{\prime }=-1, y \left (0\right )=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \cos \left (y\right ) y^{\prime }=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \cos \left (y\right ) y^{\prime }d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \sin \left (y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arcsin \left (x +c_{1} \right ) \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=2 \\ {} & {} & 2=\arcsin \left (c_{1} \right ) \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\left (\right ) \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \end {array} \]

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: 0.078 (sec). Leaf size: 13

dsolve([1=cos(y(x))*diff(y(x),x),y(0) = 2],y(x), singsol=all)
 

\[ y \left (x \right ) = \pi -\arcsin \left (x +\sin \left (2\right )\right ) \]

✗ Solution by Mathematica

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

DSolve[{1==Cos[y[x]]*y'[x],{y[0]==2}},y[x],x,IncludeSingularSolutions -> True]
 

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