4.13 problem 13

Internal problem ID [14170]
Internal file name [OUTPUT/13851_Saturday_March_02_2024_02_51_29_PM_2021726/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number: 13.
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]

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

4.13.1 Solving as quadrature ode

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

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

Summary

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

Verification of solutions

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

4.13.2 Maple step by step solution

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

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)+k*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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