# Formulating a 2nd-order ODE as a system.

• Nov 23rd 2010, 04:26 AM
bjorno
Formulating a 2nd-order ODE as a system.
I am working on an assignment, and my first task is to write this second order differential equation as a system of two first order equations. I have done this previously for special cases where f and u are linear functions (typically f(u') = b*u' and s(u) = ku), but I have no idea of what to do with this generalized equation. Can somebody help me?

mu ̈+f(u ̇)+s(u)=F(t), t>0, u(0)=U0, u ̇(0)=V0
• Nov 23rd 2010, 05:06 AM
Ackbeet
Can you please rewrite your equation? I can't read some of the symbols. They come out as gobbledy-gook.
• Nov 23rd 2010, 05:34 AM
bjorno
mu'' + f(u') + s(u) = F(t) , t > 0, u(0) = U0, u'(0) = V0
• Nov 23rd 2010, 06:09 AM
Ackbeet
The same basic procedure works. You let $y_{1}=u, y_{2}=u'.$

Then, you can say that $y_{1}'=y_{2},$ and $y_{2}'=?$

If you fill in the question mark there from the original DE, you'll be done with this step.
• Nov 23rd 2010, 06:46 AM
bjorno
is y'2 = (F(t) - f(u') - s(u))/m then?
• Nov 23rd 2010, 06:50 AM
Ackbeet
Well, it's really

$y_{2}'=\dfrac{F(t)-f(u')-s(u)}{m},$ with the prime on $y_{2},$

but then you can substitute in for $u$ and $u',$ right? What do you get? (No $u$'s allowed in final answer!)
• Nov 23rd 2010, 07:14 AM
bjorno
Since u = y1 and u' = y2, I get: (F(t) - f(y2) - s(y1)) / m
• Nov 23rd 2010, 07:15 AM
Ackbeet
So writing out your system in full gives you what?