Solved

System of first order ODE's

Posted on 2007-12-04
1
611 Views
Last Modified: 2012-05-05
So a previous question of mine was how to solve the following, by first converting the two first orders into a single second order:

dx/dt = 4x - y
dy/dt = 2x + y + t^2                                  (x(0)=0, y(0)=1)

BigRat pointed out the rather trivial method:

d^2(x)/dt^2 = 4dx/dt - dy/dt
                    = 4 dx/dt - (2x + y + t^2)
                    = 4 dx/dt - 2x - (4x - dx/dt) - t^2
=>
x'' - 5x' +6x = -t^2


Now, solving this, gives:

x = Ae^(2t) + Be^(3t) - 1/6*t^2 - 5/18*t - 19/108       (note: two unknowns)

If I apply x(0)=0:

0 = A + B - 19/108.                 (*)

Now, to solve for y (and substitute in my solution for x(t), from above):

dy/dt = 2x + y + t^2
         = 2(Ae^(2t) + Be^(3t) - 1/6*t^2 - 5/18*t - 19/108) + y + t^2
=>
d/dt(y*e^(-t)) = e^(-t) * {2(Ae^(2t) + Be^(3t) - 1/6*t^2 - 5/18*t - 19/108) + t^2}
=>
y = e^t * Integral { e^(-t) * {2(Ae^(2t) + Be^(3t) - 1/6*t^2 - 5/18*t - 19/108) + t^2} } dt
=>
y = 2Ae^(2t) + Be^(3t) - 1/54*e^(-t)*(23 + 42t + 36t^2) + Ce^t     (3 unknowns)


I can then apply y(0)=1:

1 = 2A + B - 23/54 + C.         (**)

So I have two equations, and three unknowns.... Have I lost information somewhere? Is it possible to eliminate all unknown constants?? I can't see how, but the question seems to imply that this is what is expected.....


Thanks for any input
0
Comment
Question by:InteractiveMind
1 Comment
 
LVL 3

Accepted Solution

by:
Dimkov earned 500 total points
ID: 20405400
I think you are fine
C can be any number you want to be.
Here is my logic:
x=...+C
dx/dt=... without C

dx/dt will be fine for any set of functions x=... +C

I am not sure about this. It is just an idea
0

Featured Post

Problems using Powershell and Active Directory?

Managing Active Directory does not always have to be complicated.  If you are spending more time trying instead of doing, then it's time to look at something else. For nearly 20 years, AD admins around the world have used one tool for day-to-day AD management: Hyena. Discover why

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Suggested Solutions

Title # Comments Views Activity
Two Dice Roll Probabilities 3 65
Odds of picking games correctly 4 92
Currency Conversion? 1 87
Request to review costing formula 3 36
Complex Numbers are funny things.  Many people have a basic understanding of them, some a more advanced.  The confusion usually arises when that pesky i (or j for Electrical Engineers) appears and understanding the meaning of a square root of a nega…
Foreword (May 2015) This web page has appeared at Google.  It's definitely worth considering! https://www.google.com/about/careers/students/guide-to-technical-development.html How to Know You are Making a Difference at EE In August, 2013, one …
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…

821 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question