1.11 problem 5(a)

Internal problem ID [5922]
Internal file name [OUTPUT/5170_Sunday_June_05_2022_03_26_42_PM_62334749/index.tex]

Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section: Chapter 1.3 Introduction– Linear equations of First Order. Page 38
Problem number: 5(a).
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]

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

1.11.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{y k}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 +x k}\\ &=c_{1} {\mathrm e}^{x k} \end {align*}

Summary

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

Verification of solutions

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

1.11.2 Maple step by step solution

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

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: 10

dsolve(diff(y(x),x)=k*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 18

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

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