# Representing DE as systems

• Jan 15th 2012, 06:25 AM
jonnyl
Representing DE as systems
Here is my revision question. I dont know where to start. Can someone please show me how to do this?
• Jan 15th 2012, 08:35 AM
ILikeSerena
Re: Representing DE as systems
Welcome to MHF, jonnyl! :)

A system of the form $\displaystyle \ddot x = g(\dot x, x, t)$ can be rewritten with $\displaystyle x_1=x$ and $\displaystyle x_2=\dot x$.

From this you get the system $\displaystyle \left[ \begin{array}{lll}\dot x_1 &=& x_2 \\ \dot x_2 &=& g(x_2, x_1, t) \end{array} \right.$
• Jan 16th 2012, 04:47 AM
jonnyl
Re: Representing DE as systems
So how would i represent this as a 3 dimensional system?
• Jan 16th 2012, 10:23 AM
ILikeSerena
Re: Representing DE as systems
Quote:

Originally Posted by jonnyl
So how would i represent this as a 3 dimensional system?

To convert the ODE to an autonomous system, you need to eliminate the dependency on the time variable t.
Introduce $\displaystyle x_3=\cos \omega t$.
Now find $\displaystyle \dot x_3$ by using only $\displaystyle x_1,x_2,x_3$. Can you?