Page 21 - LAPLACE TRANSFORM

2.0 PROPERTIES OF LAPLACE TRANSFORM

There are some useful properties of Laplace Transform than can be exploited . They
allow us to find the Laplace Transform of more difficult functions. The properties are

A. THEOREM 1 : LINEARITY

B. THEOREM 2 : THE FIRST SHIFTING

C. THEOREM 3 :THE MULTIPLICATION BY t n

2.1 THEOREM 1 : LINEARITY

Let f and f be two function of “a” and let “ b” be constant which may be negative . By using
2
1
the linearity properties and the Table of Laplace Transform, we can find the Laplace Transform
of more complicated functions.

L[af 1 ) (t + bf 2 (t )] = aL [ f 1 (t )] bL+ [ f 2 (t )]

L[af 1 ) (t + bf 2 (t )] = aL [ f 1 (t )] bL+ [ f 2 (t )]
∞
) bf
= ∫ e − st [af 1 (t + 2 (t )]
0
∞ ∞
= a ∫ e − st f 1 (t )dt + b ∫ e − st f 2 (t )dt

0 0
) bf
= aLf 1 (t + 2 (t )
Proof:

∞
) bf
= ∫ e − st [af 1 (t + 2 (t )]
0
∞ ∞
= a ∫ e − st f 1 (t )dt + b ∫ e − st f 2 (t )dt
0 0
= aLf (t + (t )
) bf
1 2

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