139k views
5 votes
Use the Laplace transform integral to prove that for any real constant k,L(kx(t))=kX(s)

1 Answer

4 votes

Final answer:

To prove L(kx(t)) = kX(s), start with the definition of the Laplace transform and apply the constant multiple rule, factoring out the constant k from the integral, resulting in the desired relationship.

Step-by-step explanation:

The student is asking how to prove that the Laplace transform of a scaled function kx(t) is equal to k times the Laplace transform of the function x(t), which is denoted kX(s). The proof involves using the definition of the Laplace transform which is an integral transform that takes a function of a real variable t (the time domain) to a function of a complex variable s (the frequency domain).

To prove this, consider the Laplace transform integral of the function kx(t) given by:

L(kx(t)) = ∫_{0}^{∞} e^{-st}kx(t) dt

Since k is a constant, it can be factored out of the integral:

L(kx(t)) = k ∫_{0}^{∞} e^{-st}x(t) dt

This is k times the Laplace transform of x(t), thus we get:

L(kx(t)) = kX(s)

Where X(s) is the Laplace transform of the function x(t).

User ERaufi
by
8.1k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories