2.1 problem 1

Internal problem ID [11357]
Internal file name [OUTPUT/10340_Wednesday_May_17_2023_07_49_26_PM_49721640/index.tex]

Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.1.3 Geometric. Exercises page 15
Problem number: 1.
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]

\[ \boxed {x^{\prime }-x \left (1-\frac {x}{4}\right )=0} \]

2.1.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int -\frac {4}{x \left (-4+x \right )}d x &= t +c_{1}\\ -\ln \left (-4+x \right )+\ln \left (x \right )&=t +c_{1} \end {align*}

Solving for \(x\) gives these solutions \begin {align*} x_1&=\frac {4 \,{\mathrm e}^{t +c_{1}}}{-1+{\mathrm e}^{t +c_{1}}}\\ &=\frac {4 \,{\mathrm e}^{t} c_{1}}{-1+{\mathrm e}^{t} c_{1}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {4 \,{\mathrm e}^{t} c_{1}}{-1+{\mathrm e}^{t} c_{1}} \\ \end{align*}

Figure 25: Slope field plot

Verification of solutions

\[ x = \frac {4 \,{\mathrm e}^{t} c_{1}}{-1+{\mathrm e}^{t} c_{1}} \] Verified OK.

2.1.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-x \left (1-\frac {x}{4}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=x \left (1-\frac {x}{4}\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{x \left (1-\frac {x}{4}\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{x \left (1-\frac {x}{4}\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (-4+x\right )+\ln \left (x\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=\frac {4 \,{\mathrm e}^{t +c_{1}}}{-1+{\mathrm e}^{t +c_{1}}} \end {array} \]

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: 0.016 (sec). Leaf size: 17

dsolve(diff(x(t),t)=x(t)*(1-x(t)/4),x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {4}{1+4 \,{\mathrm e}^{-t} c_{1}} \]

✓ Solution by Mathematica

Time used: 0.439 (sec). Leaf size: 32

DSolve[x'[t]==x[t]*(1-x[t]/4),x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {4 e^t}{e^t+e^{4 c_1}} \\ x(t)\to 0 \\ x(t)\to 4 \\ \end{align*}