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## Homework Statement

We have a driven pendulum described by the following differential equation:

[itex] \frac{d^2\theta}{dt^2} = \frac{-g}{l}\sin(\theta) + C\cos(\theta)\sin(\Omega t) [/itex]

I need to turn this second order differential equation into a system of first order differential equations (then use a computer to solve the first orders, but that's not the problem here).

## Homework Equations

None needed

## The Attempt at a Solution

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We are told to use some numeric values: l = 10cm, g = 9.81m/s^2, capital omega = 5/s, C = 2/s^2, and we are told to turn the equation into a dimensionless equation using the following notation:

[itex] \omega^2 = g/l [/itex]

[itex] \beta = \frac{\Omega}{\omega} [/itex]

[itex] \gamma = \frac{C}{\omega} [/itex]

[itex] x= \omega t [/itex]

Now, putting these into the ODE gives

[itex] \frac{d^2\theta}{dt^2} = \omega^2\sin(\theta) + \omega^2\gamma\cos(\theta)\sin(\beta x) [/itex]

But, the only way I can think of turning this into a system of first order ODEs is by using some dummy variable, y.

In other words, let

[itex] \frac{d\theta}{dt} = y [/itex]

and

[itex] \frac{dy}{dt} = \omega^2\sin(\theta) + \omega^2\gamma\cos(\theta)\sin(\beta x) [/itex]

Is there no way to get it all in terms of theta and x?

**EDIT:**

I accidentally posted this before it was complete because I hit the "enter" key. Is there a way to turn this feature off? I don't want to get into trouble over posting something which doesn't fit with the rules.

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