2.28 problem 40

Internal problem ID [4977]
Internal file name [OUTPUT/4470_Sunday_June_05_2022_02_57_20_PM_31826978/index.tex]

Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section: Chapter 2, First order differential equations. Section 2.3, Linear equations. Exercises. page 54
Problem number: 40.
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]

\[ \boxed {u^{\prime }-\alpha \left (1-u\right )+\beta u=0} \]

2.28.1 Solving as quadrature ode

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

Solving for \(u\) gives these solutions \begin {align*} \end {align*}

Summary

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

Verification of solutions

\[ u = \frac {{\mathrm e}^{-\alpha c_{1} -t \alpha -\beta c_{1} -t \beta }+\alpha }{\alpha +\beta } \] Verified OK.

2.28.2 Maple step by step solution

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

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: 0.016 (sec). Leaf size: 24

dsolve(diff(u(t),t)=alpha*(1-u(t))-beta*u(t),u(t), singsol=all)
 

\[ u \left (t \right ) = \frac {c_{1} \left (\alpha +\beta \right ) {\mathrm e}^{-\left (\alpha +\beta \right ) t}+\alpha }{\alpha +\beta } \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 35

DSolve[u'[t]==\[Alpha]*(1-u[t])-\[Beta]*u[t],u[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} u(t)\to \frac {\alpha }{\alpha +\beta }+c_1 e^{-t (\alpha +\beta )} \\ u(t)\to \frac {\alpha }{\alpha +\beta } \\ \end{align*}