Ordinary Differential Equations/Laplace Transform

Definition Let be a function on . The Laplace transform of is defined by the integral The domain of is all values of such that the integral exists. Existence Properties Linearity Let and be functions whose Laplace transforms exist for and let and be constants. Then, for , which can be proved using the properties of improper integrals. Shifting in s If the Laplace transform exists for , then for . Proof. Laplace Transform of Higher-Order Derivatives If , then Proof: (integrating by parts) Using the above and the linearity of Laplace Transforms, it is easy to prove that Derivatives of the Laplace Transform If , then Laplace Transform of Few Simple Functions Inverse Laplace Transform Definition Linearity