# Converting a differential equation into its Laplace transform

Given the differential equation of a current controlled electric actuator, how would I convert the differential equation into its Laplace transform?

$$J \frac{d^2 \phi(t)}{dt^2} + D \frac{d \phi(t)}{dt} + H \phi(t) = K i(t) = 0$$

I know the reason you're meant to convert this to laplace is because it's easier to work with but I'm finding it difficult to understand why you do certain things in certain parts.

Any concrete method on how to convert this would be appreciated.

If J, D, H, and K are independent of time, you use the property of the Laplace transform that the derivative in the time domain is the same as multiplying by $s$ in the frequency domain. Then you have to subtract off the initial condition as shown in the first two examples on these Cal State Fullerton math pages. But really, any web search for differentiation and Laplace transform will get you multiple hits that explain it.
$$J (s^2 \Phi(s) - s \phi(0) - \phi\prime(0)) + D (s \Phi(s) - \phi(0)) + H \Phi(s) = K I(s)$$
• You are using the original function $\phi$ here instead of its Laplace Transform $\Phi$ here .. – CrepusculeWithNellie Dec 22 '16 at 3:00