Determine a function f(t) that has the given Laplace transform F(s). F(s)=(4 s+5)/(s^2+9)

Name: Problem 21, Chapter 10: Differential Equations and Linear Algebra
Uploaded: 2020-07-02T02:58:28-08:00
Duration: 1 min 46 s
Channel: Ryan Williams
Description: Determine a function f(t) that has the given Laplace transform F(s). F(s)=(4 s+5)/(s^2+9)

step 1 So our first step here is going to be regrading on capital F of S as two separate fractions. One

Determine a function f(t) that has the given Laplace...

Key Concepts

Laplace Transform

The Laplace transform is an integral transform that converts a time-domain function into a complex frequency-domain function. This transformation simplifies the handling of differential equations by converting differentiation into algebraic operations, making it a powerful tool in both engineering and mathematics.

Inverse Laplace Transform

The inverse Laplace transform is the process used to recover the original time-domain function from its Laplace transform. It typically involves matching parts of the transform to known standard forms from Laplace transform tables and applying techniques such as partial fraction decomposition when necessary.

Linearity of Laplace Transforms

The linearity property of the Laplace transform states that the transform of a sum of functions is equal to the sum of their individual transforms. This characteristic allows for the decomposition of complex expressions into simpler components, which can be individually transformed or inverted before reassembling the final result.

Standard Transforms for Sine and Cosine

Many Laplace transforms are built upon standard forms, such as those for sine and cosine functions. The transform of a cosine function generally appears with a numerator of s and that of a sine function with the frequency multiplier. Recognizing these forms in a given Laplace transform can significantly simplify the inversion process.

Decomposition of Rational Functions

Decomposing a rational function in the Laplace domain involves breaking the expression into simpler fractions that correspond to known Laplace transform pairs. This decomposition is essential for identifying components that match standard inverse transforms, allowing for the systematic construction of the time-domain function.

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