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).