4.14 problem 16

Internal problem ID [971]
Internal file name [OUTPUT/971_Sunday_June_05_2022_01_55_38_AM_93564178/index.tex]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 2, First order equations. Existence and Uniqueness of Solutions of Nonlinear Equations. Section 2.3 Page 60
Problem number: 16.
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]

\[ \boxed {y^{\prime }-y^{\frac {2}{5}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}

4.14.1 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 {2}{5}} \end {align*}

The \(y\) domain of \(f(x,y)\) when \(x=0\) is \[ \{0\le y\} \] And the point \(y_0 = 1\) 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 {2}{5}}\right ) \\ &= \frac {2}{5 y^{\frac {3}{5}}} \end {align*}

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

4.14.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{y^{\frac {2}{5}}}d y &= \int {dx}\\ \frac {5 y^{\frac {3}{5}}}{3}&= x +c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=1\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} {\frac {5}{3}} = c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = {\frac {5}{3}} \end {align*}

Trying the constant \begin {align*} c_{1} = {\frac {5}{3}} \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} \frac {5 y^{\frac {3}{5}}}{3} = x +\frac {5}{3} \end {align*}

The constant \(c_{1} = {\frac {5}{3}}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} \frac {5 y^{\frac {3}{5}}}{3} &= x +\frac {5}{3} \\ \end{align*}

Verification of solutions

\[ \frac {5 y^{\frac {3}{5}}}{3} = x +\frac {5}{3} \] Verified OK.

4.14.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-y^{\frac {2}{5}}=0, y \left (0\right )=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{\frac {2}{5}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{\frac {2}{5}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{\frac {2}{5}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {5 y^{\frac {3}{5}}}{3}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\left (\frac {3 x}{5}+\frac {3 c_{1}}{5}\right )^{\frac {5}{3}} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=\frac {3 \,3^{\frac {2}{3}} 5^{\frac {1}{3}} c_{1}^{\frac {5}{3}}}{25} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\frac {5}{3} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\frac {5}{3}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {\left (3 x +5\right ) \left (\frac {3 x}{5}+1\right )^{\frac {2}{3}}}{5} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\left (3 x +5\right ) \left (\frac {3 x}{5}+1\right )^{\frac {2}{3}}}{5} \end {array} \]

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: 0.031 (sec). Leaf size: 11

dsolve([diff(y(x),x)=y(x)^(2/5),y(0) = 1],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (3 x +5\right ) \left (\frac {3 x}{5}+1\right )^{\frac {2}{3}}}{5} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 23

DSolve[{y'[x]==y[x]^(2/5),y[0]==1},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {(3 x+5)^{5/3}}{5\ 5^{2/3}} \]