2.12 problem 14

Internal problem ID [22]
Internal file name [OUTPUT/22_Sunday_June_05_2022_01_33_50_AM_29491422/index.tex]

Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.3. Slope fields and solution curves. Page 26
Problem number: 14.
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]

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

2.12.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= y^{\frac {1}{3}} \end {align*}

The \(y\) domain of \(f(x,y)\) when \(x=0\) 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 (y^{\frac {1}{3}}\right ) \\ &= \frac {1}{3 y^{\frac {2}{3}}} \end {align*}

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

2.12.2 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.

2.12.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-y^{\frac {1}{3}}=0, y \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{\frac {1}{3}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{\frac {1}{3}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{\frac {1}{3}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {3 y^{\frac {2}{3}}}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (6 x +6 c_{1} \right )^{\frac {3}{2}}}{27} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=\frac {2 \sqrt {6}\, c_{1}^{\frac {3}{2}}}{9} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {2 \sqrt {6}\, x^{\frac {3}{2}}}{9} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {2 \sqrt {6}\, x^{\frac {3}{2}}}{9} \end {array} \]

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: 0.0 (sec). Leaf size: 5

dsolve([diff(y(x),x) = y(x)^(1/3),y(0) = 0],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 21

DSolve[{y'[x] == y[x]^(1/3),y[0]==0},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {2}{3} \sqrt {\frac {2}{3}} x^{3/2} \]