1.58 problem 80

Internal problem ID [14101]
Internal file name [OUTPUT/13782_Saturday_March_02_2024_02_50_31_PM_47259356/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 80.
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]

\[ \boxed {y^{\prime }=\sin \left (x \right )^{4}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}

1.58.1 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) &=\sin \left (x \right )^{4} \end {align*}

Hence the ode is \begin {align*} y^{\prime } = \sin \left (x \right )^{4} \end {align*}

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

1.58.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \sin \left (x \right )^{4}\,\mathop {\mathrm {d}x}}\\ &= \frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4}+c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=0\) 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*} y&=\frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4} \end {align*}

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

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4} \] Verified OK.

1.58.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\sin \left (x \right )^{4}, y \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \sin \left (x \right )^{4}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {\sin \left (x \right )^{3} \cos \left (x \right )}{4}-\frac {3 \sin \left (x \right ) \cos \left (x \right )}{8}+\frac {3 x}{8}+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {\left (2 \cos \left (x \right )^{3}-5 \cos \left (x \right )\right ) \sin \left (x \right )}{8}+\frac {3 x}{8} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\left (2 \cos \left (x \right )^{3}-5 \cos \left (x \right )\right ) \sin \left (x \right )}{8}+\frac {3 x}{8} \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 20

dsolve([diff(y(x),x)=sin(x)^4,y(0) = 0],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 23

DSolve[{y'[x]==Sin[x]^4,{y[0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{32} (12 x-8 \sin (2 x)+\sin (4 x)) \]