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
October 11th, 2010, 06:37 PM   #1
Member
 
Joined: Sep 2010

Posts: 73
Thanks: 0

you are given general solution - find differential equation

You are given the general solution - find the homogeneous 2nd order differential equation ay'' + by' + cy = 0.
y(x) = c_1 + c_2 * e^(-10x)

So I got y'(x) = -10 c_2 e^(-10x) and y''(x) = 100 c_2 e^(-10x)

And I see from the given general solution that e^(0x) = 1 and e^(-10x) are two particular solutions, y_1 and y_2.
So given that the book says we are supposed to guess y_1 = e^(r_1 * x) and y_2 = e^(r_2 * x) [I still don't understand WHY we have to 'guess' this],
that means my r_1 = 0 and my r_2 = -10.

Now.. how do I find the differential equation from here?
mbradar2 is offline  
 
October 11th, 2010, 07:27 PM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 20,931
Thanks: 2205

y'' + 10y' = 0
skipjack is online now  
October 11th, 2010, 07:47 PM   #3
Member
 
Joined: Sep 2010

Posts: 73
Thanks: 0

Re: you are given general solution - find differential equat

Yes, I know that's the answer, but HOW do I get there from where I am?
mbradar2 is offline  
October 11th, 2010, 08:18 PM   #4
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Re: you are given general solution - find differential equat

You are told your auxiliary equation has roots 0 and -10, thus

rČ + 10r = 0

Since e^(rx) cannot be zero, multiply through.

(rČ + 10r)e^(rx) = 0

(a) rČe^(rx) + 10re^(rx) = 0

We are given y = c1 + c2e^(rx), thus

y' = c2re^(rx)
y'' = c2rČe^(rx)

Thus, substituting into (a), we may state:

y''/c2 + 10y'/c2 = 0

Multiply through by c2.

y'' + 10y' = 0
MarkFL is offline  
October 12th, 2010, 02:10 AM   #5
Global Moderator
 
Joined: Dec 2006

Posts: 20,931
Thanks: 2205

You can go directly from the auxiliary equation to your answer (or vice versa). Up to multiplication by a constant (and rearrangement of terms), the result is unique. [color=#00AA00]MarkFL[/color] was aiming for a rigorous proof, but overlooked the possibility that c2 = 0.
skipjack is online now  
October 12th, 2010, 09:20 AM   #6
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Re: you are given general solution - find differential equat

Yes, I did make the assumption that c2 ? 0 thereby missing the trivial solution y(x) = c1. I was just trying to show a path to the non-trivial solution given by [color=#00BF00]skipjack[/color].
MarkFL is offline  
October 12th, 2010, 10:29 AM   #7
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Re: you are given general solution - find differential equat

Lol, I really should get my brain in gear before typing away. I ignored, rather than missed the trivial solution. And the equation given encompasses all solutions, trivial or otherwise. In other words, ignore what I said.
MarkFL is offline  
October 12th, 2010, 01:09 PM   #8
Global Moderator
 
Joined: Dec 2006

Posts: 20,931
Thanks: 2205

Multiplying rČ + 10r = 0 by c2e^(rx) gives c2e^(rx)rČ + 10c2e^(rx)r = 0 (whether or not c2e^(rx) = 0).
Now making the substitutions you suggested gives y'' + 10y' = 0.
skipjack is online now  
Reply

  My Math Forum > College Math Forum > Differential Equations

Tags
differential, equation, find, general, solution



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Differential Equations: General & Particular Solution aliaja Differential Equations 1 September 17th, 2013 03:31 PM
Finding the general solution of this differential equation dunn Differential Equations 2 February 19th, 2012 03:38 AM
differential equation: find general solution mbradar2 Differential Equations 7 September 25th, 2010 10:58 AM
unable to find solution for a differential equation geeth416 Differential Equations 2 December 9th, 2009 12:57 AM
the general solution of a homogeous differential equation. Ichigo Differential Equations 3 September 1st, 2008 10:06 AM





Copyright © 2019 My Math Forum. All rights reserved.