# Driven Damped Harmonic Oscillator, f != ma?

Driven Damped Harmonic Oscillator, f != ma??

Let's say I've got a driven damped harmonic oscillator described by the following equation:
$$A \ddot{x} + B \dot{x} + C x = D f(t)$$

given that $$f = ma$$ why can't I write

$$A \ddot{x} + B \dot{x} + C x = D ma$$

substitute $$\ddot{x} = a$$ to get

$$A \ddot{x} + B \dot{x} + C x = D m \ddot{x}$$

and then rearrange to get

$$(A - D m) \ddot{x} + B \dot{x} + C x = 0$$

I know that's not how the problem is solved, but what is to stop me from solving it that way?

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## Answers and Replies

Are you assuming constant acceleration?

You're forgetting that f(t) is a time-varying force. You also have forces that depend on velocity and position, which is why what you wrote isn't correct. You only got the time-varying force.

Meir Achuz
$$A \ddot{x} + B \dot{x} + C x = D f(t)$$
given that $$f = ma$$ why can't I write