5.24 problem 9

Internal problem ID [11421]
Internal file name [OUTPUT/10404_Thursday_May_18_2023_04_18_30_AM_37563748/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 1.4.1. Integrating factors. Exercises page 41
Problem number: 9.
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]

\[ \boxed {x^{\prime }-a x=b} \]

5.24.1 Solving as quadrature ode

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

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

Summary

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

Verification of solutions

\[ x = \frac {c_{1} {\mathrm e}^{t a}-b}{a} \] Verified OK.

5.24.2 Maple step by step solution

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

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: 19

dsolve(diff(x(t),t)=a*x(t)+b,x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {{\mathrm e}^{a t} c_{1} a -b}{a} \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 30

DSolve[x'[t]==a*x[t]+b,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {b}{a}+c_1 e^{a t} \\ x(t)\to -\frac {b}{a} \\ \end{align*}