My Math Forum  

Go Back   My Math Forum > College Math Forum > Differential Equations

Differential Equations Ordinary and Partial Differential Equations Math Forum


Reply
 
LinkBack Thread Tools Display Modes
February 19th, 2017, 05:08 AM   #1
Newbie
 
Joined: Feb 2017
From: Karachi,Pakistan

Posts: 1
Thanks: 0

Linear ordinary differential equation

Here is an equation
Vsinwt=iR+Ldi/dt
i have to solve this equation with linear method ,i.e i'+p(t)i=Q(t) i have solved it to the step Li'e^rt+iRe^rt=e^rtVsinwt
by initial steps and by finding integrating factor ,but now i stuck ,just don't know that what to do with that L with first most term,because if there is no L,there is a formula of d/dx(u.v) on L.h.s and it will proceed for solution of equation. .so can you plz tell me that what to do with that L in the step where i reached or from the main equation given?
mnizami is offline  
 
February 19th, 2017, 01:24 PM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: CA

Posts: 914
Thanks: 494

You won't solve this using an integrating factor.

You need to first solve the homogeneous equation

$R i + L i^\prime = 0$

and then use the method of undetermined coefficients to solve for the particular solution given the driving function $V \sin(\omega t)$

The final solution is the sum of the homogeneous solution and the particular solution.

With a sinusoidal driving function you should try a function of the form

$i_p(t) = A \cos(\omega t) + B \sin(\omega t)$

plug that into the differential equation and solve for the $A, ~B$ that makes that equal to the driving function $V \sin(\omega t)$

It's a bit of differentiation and a bunch of algebra.

I have confidence you can do it!
romsek is offline  
February 19th, 2017, 03:45 PM   #3
Member
 
Joined: Oct 2016
From: Melbourne

Posts: 77
Thanks: 35

You CAN use the integrating factor method, the IV is "t" and the DV is "i" and all other letters are constants.

$\displaystyle \begin{align*} V\sin{ \left( \omega \, t \right) } &= i\,R + L\,\frac{\mathrm{d}i}{\mathrm{d}t} \\ \frac{\mathrm{d}i}{\mathrm{d}t} + \frac{R}{L}\,i &= \frac{V}{L}\,\sin{ \left( \omega\,t \right) } \end{align*}$

The integrating factor is $\displaystyle \begin{align*} \mathrm{e}^{\int{ \frac{R}{L}\,\mathrm{d}t }} = \mathrm{e}^{\frac{R}{L}\,t} \end{align*}$ so multiplying both sides of the equation by this gives

$\displaystyle \begin{align*} \mathrm{e}^{\frac{R}{L}\,t}\,\frac{\mathrm{d}i}{ \mathrm{d} t} + \frac{R}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,i &= \frac{V}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,\sin{ \left( \omega\,t \right) } \\ \frac{\mathrm{d}}{\mathrm{d}t}\,\left( \mathrm{e}^{\frac{R}{L}\,t}\,i \right) &= \frac{V}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,\sin{ \left( \omega\,t \right) } \\ \mathrm{e}^{\frac{R}{L}\,t}\,i &= \int{ \frac{V}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,\sin{ \left( \omega\,t \right) }\,\mathrm{d}t } \end{align*}$

You can now perform the integration using integration by parts twice.
Prove It is offline  
February 19th, 2017, 05:45 PM   #4
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 6,445
Thanks: 2115

Math Focus: Mainly analysis and algebra
The OP has missed that in the form $$i' + p(t)i = Q(t)$$
the coefficient of $i'$ is 1. This essential for the method of the integrating factor to work.
v8archie is offline  
Reply

  My Math Forum > College Math Forum > Differential Equations

Tags
differential, equation, linear, ordinary



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Linear Ordinary Differential Equation with Constant Coefficients zylo Calculus 5 November 30th, 2016 07:56 PM
Ordinary Differential Equation Abdus Salam Differential Equations 1 April 28th, 2015 06:39 PM
ordinary differential equation aheed Differential Equations 3 February 22nd, 2014 08:03 AM
Ordinary differential equation y' = 2y + 4 Norm850 Differential Equations 2 February 1st, 2012 12:57 AM
an ordinary differential equation allison711 Differential Equations 3 February 8th, 2008 09:19 AM





Copyright © 2017 My Math Forum. All rights reserved.