1.2 problem 2.2 (b)

Internal problem ID [13243]
Internal file name [OUTPUT/12415_Wednesday_February_14_2024_02_06_05_AM_81956848/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: 2.2 (b).
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]

\[ \boxed {y^{\prime }+\sin \left (y\right )=3} \]

1.2.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{3-\sin \left (y \right )}d y &= x +c_{1}\\ \frac {\sqrt {2}\, \arctan \left (\frac {\left (6 \tan \left (\frac {y}{2}\right )-2\right ) \sqrt {2}}{8}\right )}{2}&=x +c_{1} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=2 \arctan \left (\frac {\left (\sqrt {2}+4 \tan \left (\left (x +c_{1} \right ) \sqrt {2}\right )\right ) \sqrt {2}}{6}\right ) \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \arctan \left (\frac {\left (\sqrt {2}+4 \tan \left (\left (x +c_{1} \right ) \sqrt {2}\right )\right ) \sqrt {2}}{6}\right ) \\ \end{align*}

Figure 2: Slope field plot

Verification of solutions

\[ y = 2 \arctan \left (\frac {\left (\sqrt {2}+4 \tan \left (\left (x +c_{1} \right ) \sqrt {2}\right )\right ) \sqrt {2}}{6}\right ) \] Verified OK.

1.2.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\sin \left (y\right )=3 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=3-\sin \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{3-\sin \left (y\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{3-\sin \left (y\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\sqrt {2}\, \arctan \left (\frac {\left (6 \tan \left (\frac {y}{2}\right )-2\right ) \sqrt {2}}{8}\right )}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 \arctan \left (\frac {\left (\sqrt {2}+4 \tan \left (\left (x +c_{1} \right ) \sqrt {2}\right )\right ) \sqrt {2}}{6}\right ) \end {array} \]

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: 0.016 (sec). Leaf size: 22

dsolve(diff(y(x),x)=3-sin(y(x)),y(x), singsol=all)
 

\[ y \left (x \right ) = 2 \arctan \left (\frac {1}{3}+\frac {2 \sqrt {2}\, \tan \left (\left (c_{1} +x \right ) \sqrt {2}\right )}{3}\right ) \]

✓ Solution by Mathematica

Time used: 5.504 (sec). Leaf size: 83

DSolve[y'[x]==3-Sin[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -2 \arctan \left (\frac {1}{3} \left (-1-2 \sqrt {2} \tan \left (\sqrt {2} (x-c_1)\right )\right )\right ) \\ y(x)\to 2 \arctan \left (\frac {1}{3} \left (1+2 \sqrt {2} \tan \left (\sqrt {2} (x-c_1)\right )\right )\right ) \\ y(x)\to \arcsin (3) \\ y(x)\to \text {Interval}[\{-\pi ,\pi \}] \\ \end{align*}