If you have a function , with corresponding Laplace transform , then the Laplace transform of is , where the upper- and lower-case letters are very carefully typed. Similarly, the second derivative of would have the Laplace transform . So you gain two things by Laplace transformation: differentiation becomes multiplication, and the initial conditions are automatically included in your answer! Of course, not all functions evenhavea Laplace transform, so this method can only get you so far.