5.2 problem Exercise 11.2, page 97

Internal problem ID [4496]
Internal file name [OUTPUT/3989_Sunday_June_05_2022_11_59_40_AM_1736555/index.tex]

Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 11, Bernoulli Equations
Problem number: Exercise 11.2, page 97.
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]

\[ \boxed {y^{\prime }+a y=b} \]

5.2.1 Solving as quadrature ode

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

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

Summary

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

Verification of solutions

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

5.2.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+a y=b \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-a y+b \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-a y+b}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-a y+b}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-a y+b \right )}{a}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {{\mathrm e}^{-c_{1} a -a x}-b}{a} \end {array} \]

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: 18

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

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

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 29

DSolve[y'[x]+a*y[x]==b,y[x],x,IncludeSingularSolutions -> True]
 

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