1.27 problem 2.4 (e)

Internal problem ID [13268]
Internal file name [OUTPUT/12440_Wednesday_February_14_2024_02_06_16_AM_67654846/index.tex]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page 32
Problem number: 2.4 (e).
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]

\[ \boxed {\cos \left (x \right ) y^{\prime }=\sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}

1.27.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}

Where here \begin {align*} p(x) &=0\\ q(x) &=\frac {\sin \left (x \right )}{\cos \left (x \right )} \end {align*}

Hence the ode is \begin {align*} y^{\prime } = \frac {\sin \left (x \right )}{\cos \left (x \right )} \end {align*}

The domain of \(p(x)=0\) is \[ \{-\infty

1.27.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \frac {\sin \left (x \right )}{\cos \left (x \right )}\,\mathop {\mathrm {d}x}}\\ &= -\ln \left (\cos \left (x \right )\right )+c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=3\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 3 = c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = 3 \end {align*}

Trying the constant \begin {align*} c_{1} = 3 \end {align*}

Substituting this in the general solution gives \begin {align*} y&=-\ln \left (\cos \left (x \right )\right )+3 \end {align*}

The constant \(c_{1} = 3\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\ln \left (\cos \left (x \right )\right )+3 \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = -\ln \left (\cos \left (x \right )\right )+3 \] Verified OK.

1.27.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\cos \left (x \right ) y^{\prime }=\sin \left (x \right ), y \left (0\right )=3\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sin \left (x \right )}{\cos \left (x \right )} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sin \left (x \right )}{\cos \left (x \right )}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\ln \left (\cos \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\ln \left (\cos \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=3 \\ {} & {} & 3=c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =3 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =3\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\ln \left (\cos \left (x \right )\right )+3 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\ln \left (\cos \left (x \right )\right )+3 \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 11

dsolve([cos(x)*diff(y(x),x)-sin(x)=0,y(0) = 3],y(x), singsol=all)
 

\[ y \left (x \right ) = -\ln \left (\cos \left (x \right )\right )+3 \]

✓ Solution by Mathematica

Time used: 0.01 (sec). Leaf size: 12

DSolve[{Cos[x]*y'[x]-Sin[x]==0,{y[0]==3}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 3-\log (\cos (x)) \]