2.1 problem 1

Internal problem ID [887]
Internal file name [OUTPUT/887_Sunday_June_05_2022_01_53_19_AM_9951064/index.tex]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
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]

\[ \boxed {y^{\prime }+a y=0} \]

2.1.1 Solving as quadrature ode

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

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

Summary

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

Verification of solutions

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

2.1.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+a y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-a y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-a \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -a d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-x a +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-x a +c_{1}} \end {array} \]

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: 11

dsolve(diff(y(x),x) + a*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 19

DSolve[y'[x] + a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 e^{-a x} \\ y(x)\to 0 \\ \end{align*}