Representing DE as systems

**jonnyl** · January 15th 2012, 06:25 AM

Here is my revision question. I dont know where to start. Can someone please show me how to do this?

**ILikeSerena** · January 15th 2012, 08:35 AM

Welcome to MHF, jonnyl!

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

From this you get the system $\left[ \begin{array}{lll}\dot x_1 &=& x_2 \\ \dot x_2 &=& g(x_2, x_1, t) \end{array} \right.$

**jonnyl** · January 16th 2012, 04:47 AM

So how would i represent this as a 3 dimensional system?

**ILikeSerena** · January 16th 2012, 10:23 AM

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 $x_3=\cos \omega t$ .
Now find $\dot x_3$ by using only $x_1,x_2,x_3$ . Can you?

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