4.8 problem 1(h)

Internal problem ID [11374]
Internal file name [OUTPUT/10357_Wednesday_May_17_2023_07_49_53_PM_61469210/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 1.3.1 Separable equations. Exercises page 26
Problem number: 1(h).
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]

\[ \boxed {y^{\prime }-r \left (a -y\right )=0} \]

4.8.1 Solving as quadrature ode

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

Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-c_{1} r -t r}+a\\ &=\frac {{\mathrm e}^{-t r}}{c_{1}}+a \end {align*}

Summary

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

Verification of solutions

\[ y = \frac {{\mathrm e}^{-t r}}{c_{1}}+a \] Verified OK.

4.8.2 Maple step by step solution

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

Maple trace

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

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 13

dsolve(diff(y(t),t)=r*(a-y(t)),y(t), singsol=all)
 

\[ y \left (t \right ) = a +{\mathrm e}^{-r t} c_{1} \]

✓ Solution by Mathematica

Time used: 0.064 (sec). Leaf size: 21

DSolve[y'[t]==r*(a-y[t]),y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to a+c_1 e^{-r t} \\ y(t)\to a \\ \end{align*}