Transforms of Derivatives and Integrals ODEs.
The Laplace transform is a method of solving ODEs and initial value problems. The crucial idea is that operations of calculus on functions are replaced by operations of algebra on transforms. Roughly, differentiation of f(x) will correspond to multiplication of L(x) by s and integration of to division of by s. To solve ODEs, we must first consider the Laplace transform of derivatives. You have encountered such an idea in your study of logarithms. Under the application of the natural logarithm, a product of numbers becomes a sum of their logarithms, a division of numbers becomes their difference of logarithms. To simplify calculations was one of the main reasons that logarithms were invented in pre-computer times.
T H E O R E M 1 Laplace Transform of Derivatives
The transforms of the first and second derivatives of satisfy
(1)
(2)
Formula (1) holds if is continuous for all and satisfies the growth
restriction (2) in Sec. 6.1 and is piecewise continuous on every finite interval
on the semi-axis Similarly, (2) holds if f and are continuous for all
and satisfy the growth restriction and is piecewise continuous on every finite
interval on the semi-axis t 0.
f s
t 0. f r t 0
f r(t)
f (t) t 0
l( f s) s2l( f ) sf (0) f r(0).
l( f r) sl( f ) f (0)
f (t)
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