1.24 problem 2.4 (b)

Internal problem ID [13265]
Internal file name [OUTPUT/12437_Wednesday_February_14_2024_02_06_14_AM_52232036/index.tex]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page 32
Problem number: 2.4 (b).
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]

\[ \boxed {\left (x +6\right )^{\frac {1}{3}} y^{\prime }=1} \] With initial conditions \begin {align*} [y \left (2\right ) = 10] \end {align*}

1.24.1 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) &=\frac {1}{\left (x +6\right )^{\frac {1}{3}}} \end {align*}

Hence the ode is \begin {align*} y^{\prime } = \frac {1}{\left (x +6\right )^{\frac {1}{3}}} \end {align*}

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

1.24.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \frac {1}{\left (x +6\right )^{\frac {1}{3}}}\,\mathop {\mathrm {d}x}}\\ &= \frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=2\) and \(y=10\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 10 = c_{1} +6 \end {align*}

The solutions are \begin {align*} c_{1} = 4 \end {align*}

Trying the constant \begin {align*} c_{1} = 4 \end {align*}

Substituting this in the general solution gives \begin {align*} y&=\frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \end {align*}

The constant \(c_{1} = 4\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \] Verified OK.

1.24.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\left (x +6\right )^{\frac {1}{3}} y^{\prime }=1, y \left (2\right )=10\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{\left (x +6\right )^{\frac {1}{3}}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{\left (x +6\right )^{\frac {1}{3}}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (2\right )=10 \\ {} & {} & 10=\frac {3 \,8^{\frac {2}{3}}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =-\frac {3 \,8^{\frac {2}{3}}}{2}+10 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =-\frac {3 \,8^{\frac {2}{3}}}{2}+10\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 13

dsolve([(x+6)^(1/3)*diff(y(x),x)=1,y(2) = 10],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 18

DSolve[{(x+6)^(1/3)*y'[x]==1,{y[2]==10}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {3}{2} (x+6)^{2/3}+4 \]