2.7 problem 3

Internal problem ID [5929]
Internal file name [OUTPUT/5177_Sunday_June_05_2022_03_26_50_PM_84062245/index.tex]

Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section: Chapter 1.6 Introduction– Linear equations of First Order. Page 41
Problem number: 3.
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]

\[ \boxed {L y^{\prime }+R y=E} \]

2.7.1 Solving as quadrature ode

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

Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {-{\mathrm e}^{-\frac {R \left (x +c_{1} \right )}{L}}+E}{R}\\ &=\frac {-\frac {{\mathrm e}^{-\frac {R x}{L}}}{c_{1}}+E}{R} \end {align*}

Summary

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

Verification of solutions

\[ y = \frac {-\frac {{\mathrm e}^{-\frac {R x}{L}}}{c_{1}}+E}{R} \] Verified OK.

2.7.2 Maple step by step solution

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

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: 21

dsolve(L*diff(y(x),x)+R*y(x)=E,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.056 (sec). Leaf size: 23

DSolve[L*y'[x]+R*y[x]==E0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\text {E0}-\text {E0} e^{-\frac {R x}{L}}}{R} \]