Properties of the Laplace Transformation
The Laplace transform is linear:
This is actually a property of the integral, and is inherited by the Laplace transform.
The Laplace transform transforms first derivatives:
whereis an initial or boundary condition of
We integrate by parts.
The Laplace transform transform second derivatives:
whereandare initial or boundary conditions ofand
Again we integrate by parts (twice).
The Laplace transform transforms integrals:
Again we integrate by parts.
At the upper limit the first term on the right vanishes becauseand at the lower limit the integral in the first term on the right is zero because the upper and lower limits are equal hence
The Laplace transform obeys the convolution principle:
Changing the order of integration gives
Make the substitutionto give