Let $ f(t) $ be a function on $[0, \infty)$. The Laplace transform of $ f $ is the function $ F $ defined by the integral:\[F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt.\]Use this definition to determine the Laplace transform of the following function:\[ f(t) = t \, e^{-4t} \]The Laplace transform of $ f(t) $ is $ F(s) = $ $\boxed{\phantom{a}}$. (Type an expression using $ s $ as the variable.)It is defined for $ s > $ $\boxed{\phantom{a}}$. (Type an integer or a fraction.)

We have find Laplace transform of given function:

Let f(t) be a function on [0, 0). The Laplace transform of f is the function F defined by the integral F(s) = e -stí(t)dt. Use this definition to determine the Laplace transform of the following func -4t f(t) = te The Laplace transform of f(t) is F(s)=| | (Type an expression using s as the variable.) It is defined for s> (Type an integer or a fraction.)

Let f(t) be a function on [0, 0). The Laplace transform of f is the function F defined by the integral F(s) = e -stí(t)dt. Use this definition to determine the Laplace transform of the following func -4t f(t) = te The Laplace transform of f(t) is F(s)=| | (Type an expression using s as the variable.) It is defined for s> (Type an integer or a fraction.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let $ f(t) $ be a function on $[0, \infty)$. The Laplace transform of $ f $ is the function $ F $ defined by the integral:

\[
F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt.
\]

Use this definition to determine the Laplace transform of the following function:

\[
f(t) = t \, e^{-4t}
\]

The Laplace transform of $ f(t) $ is $ F(s) = $ $\boxed{\phantom{a}}$. (Type an expression using $ s $ as the variable.)

It is defined for $ s > $ $\boxed{\phantom{a}}$. (Type an integer or a fraction.)

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