1550078481-Ordinary_Differential_Equations__Roberts_

538 Ordinary Differential Equations

Laplace transforms

Function Transform

f(x) F(s) = .C[f(x)]

f(cx), c>O ~F(~)

c c

f' sF(s) - f(O)

f"(x) s^2 F(s) - sf(O) - f'(O)

f(n)(x), n a positive integer Sn F(s) - sn-1 f(O) - ... - f(n-l)(Q)

fox f(~) d~

F(s)

--

s

Properties of Laplace transforms

Function Transform

Linearity Property

af(x) + bg(x), a, b constants aF(s) + bG(s)

First Translation Property

eax f(x) F(s - a)

Second Translation Property

u(x - c)f(x - c), c;:::: 0 e-cs F(s)

Convolution Theorem

fox f(x - Og(~) d~ F(s)G(s)