Find the Laplace transform of the given function. f(t)=t-u1(t)(t-1), t ≥0

Name: Problem 12, Chapter 6: Elementary Differential Equations and Boundary Value Problem
Uploaded: 2021-08-21T01:43:15-08:00
Duration: 49 s
Channel: Aidan Mcnabb
Description: Find the Laplace transform of the given function. f(t)=t-u1(t)(t-1), t ≥0

step 1 Hello and welcome to Problem 12 of Chapter 6, Section 3. Here we're asked to find the Laplace tr

Find the Laplace transform of the given function....

Key Concepts

Piecewise Functions

Piecewise functions are functions that have different expressions over different intervals of the independent variable. In the context of Laplace transforms, these functions can often be rewritten using unit step functions, allowing the application of transforms uniformly across the domain. This process is essential for analyzing systems with behavior that changes at particular points in time.

Second Shifting Theorem

The second shifting theorem, also known as the time delay property of the Laplace transform, addresses the Laplace transform of functions that are shifted in time. It states that if a function is delayed by a certain amount, its Laplace transform can be expressed as an exponential factor multiplied by the Laplace transform of the original, unshifted function. This theorem is particularly useful when working with functions multiplied by unit step functions.

Unit Step (Heaviside) Function

The unit step function, often denoted as u(t-a), is used to 'turn on' a function at a specific point in time. It is defined as zero before a set time and one afterward. This function is essential for representing piecewise-defined functions in a unified mathematical expression, which can then be easily handled in each domain of definition.

Laplace Transform

The Laplace transform is an integral transform used to convert functions of time into functions of a complex variable, typically simplifying the analysis of linear time-invariant systems by transforming differential equations into algebraic equations. This is done by integrating the product of the function and an exponential decay factor over time, typically from zero to infinity.

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