3.8 problem 8

Internal problem ID [14131]
Internal file name [OUTPUT/13812_Saturday_March_02_2024_02_50_53_PM_33294898/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number: 8.
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]

\[ \boxed {y^{\prime }-6 y^{\frac {2}{3}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}

3.8.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(t,y)\\ &= 6 y^{\frac {2}{3}} \end {align*}

The \(y\) domain of \(f(t,y)\) when \(t=1\) is \[ \{0\le y\} \] And the point \(y_0 = 0\) is inside this domain. Now we will look at the continuity of \begin {align*} \frac {\partial f}{\partial y} &= \frac {\partial }{\partial y}\left (6 y^{\frac {2}{3}}\right ) \\ &= \frac {4}{y^{\frac {1}{3}}} \end {align*}

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(t=1\) is \[ \{0

3.8.2 Solving as quadrature ode

Since ode has form \(y^{\prime }= f(y)\) and initial conditions \(y = 0\) is verified to satisfy the ode, then the solution is \begin {align*} y&=y_0 \\ &=0 \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 0 \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = 0 \] Verified OK.

3.8.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-6 y^{\frac {2}{3}}=0, y \left (1\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=6 y^{\frac {2}{3}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{\frac {2}{3}}}=6 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{\frac {2}{3}}}d t =\int 6d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & 3 y^{\frac {1}{3}}=6 t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=8 t^{3}+4 t^{2} c_{1} +\frac {2}{3} t \,c_{1}^{2}+\frac {1}{27} c_{1}^{3} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (1\right )=0 \\ {} & {} & 0=8+4 c_{1} +\frac {2}{3} c_{1}^{2}+\frac {1}{27} c_{1}^{3} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\left (-6, -6, -6\right ) \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\left (-6, -6, -6\right )\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=8 \left (-1+t \right )^{3} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=8 \left (-1+t \right )^{3} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
<- Bernoulli successful`
 

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 5

dsolve([diff(y(t),t)=6*y(t)^(2/3),y(1) = 0],y(t), singsol=all)
 

\[ y \left (t \right ) = 0 \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 6

DSolve[{y'[t]==6*y[t]^(2/3),{y[1]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 0 \]