4.44 problem 44

Internal problem ID [14201]
Internal file name [OUTPUT/13882_Saturday_March_09_2024_03_56_36_PM_24676024/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: 44.
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]

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

4.44.1 Existence and uniqueness analysis

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

Where here \begin {align*} p(y) &=0\\ q(y) &=\sin \left (y \right )^{2} \end {align*}

Hence the ode is \begin {align*} x^{\prime } = \sin \left (y \right )^{2} \end {align*}

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

4.44.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} x &= \int { \sin \left (y \right )^{2}\,\mathop {\mathrm {d}y}}\\ &= \frac {y}{2}-\frac {\sin \left (2 y \right )}{4}+c_{1} \end {align*}

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

The solutions are \begin {align*} c_{1} = 0 \end {align*}

Trying the constant \begin {align*} c_{1} = 0 \end {align*}

Substituting this in the general solution gives \begin {align*} x&=\frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \end {align*}

The constant \(c_{1} = 0\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ x = \frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \] Verified OK.

4.44.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [-x^{\prime }=-\sin \left (y \right )^{2}, x \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & x^{\prime }=\sin \left (y \right )^{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} y \\ {} & {} & \int x^{\prime }d y =\int \sin \left (y \right )^{2}d y +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x=-\frac {\cos \left (y \right ) \sin \left (y \right )}{2}+\frac {y}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=-\frac {\cos \left (y \right ) \sin \left (y \right )}{2}+\frac {y}{2}+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} x \left (0\right )=0 \\ {} & {} & 0=c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & x=\frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x=\frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 14

dsolve([sin(y)^2=diff(x(y),y),x(0) = 0],x(y), singsol=all)
 

\[ x \left (y \right ) = \frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 17

DSolve[{Sin[y]^2==x'[y],{x[0]==0}},x[y],y,IncludeSingularSolutions -> True]
 

\[ x(y)\to \frac {1}{2} (y-\sin (y) \cos (y)) \]