2.6. IMPORTANT PROPERTIES OF THE LAPLACE TRANSFORM
The Laplace transform has been introduced in order to simplify several mathematical operations. These operations center upon the solution of linear differential equations. Several basic properties of the Laplace transform are given here.
A. Addition and Subtraction
If the Laplace transforms of f1(t) and f2(t) are F1(s) and F2(s), respectively, then
[f1(t) ± f2(t)] = F1(s) ± F2(s).
B. Multiplication by a Constant
If the Laplace transform of f(t) is F(s), the multiplication of the function f(t) by a constant K results in a Laplace transform KF(s).
C. Direct Transforms of Derivatives
If the Laplace transform of f(t) is F(s), the transform of the first time derivative 
(t) of f(t) is given by
where f(0+) is the initial value of f(t), evaluated as t → 0 from the positive region. The transform of the second time derivative 
(t) of f(t) is given by
where (0+) is the first derivative of f(t) evaluated at t = 0+. The Laplace transform of the ...
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