Determine a function f(t) that has the given Laplace transform F(s). F(s)=(2 s+5)/(s(s^2+4 s+20)

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

Key Concepts

Completing the Square

Completing the square is a technique to rewrite quadratic expressions in a form that reveals their structure, such as transforming s² + bs + c into (s + d)² + e. This is crucial in identifying terms that match standard Laplace transform pairs, especially those involving sinusoidal or exponential functions.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational function into a sum of simpler fractions. This is particularly useful with Laplace transforms because it allows us to handle more manageable components that directly correspond to standard inverse Laplace transforms.

Laplace Transform

The Laplace transform is an integral transform used to convert functions from the time domain into the complex frequency domain. It simplifies the process of solving differential equations by turning derivatives into algebraic terms and is a fundamental tool in engineering and physics.

Inverse Laplace Transform

The inverse Laplace transform retrieves the original time-domain function from its Laplace domain representation. It often involves techniques like partial fraction decomposition, transformation of algebraic expressions, and using tables of known transforms to simplify the inversion process.

Transcript

00:01 The first step to this problem is going to be doing partial fraction decomposition on our fraction here, of which we're going to write this as some fraction a over s, plus b, s plus c, all over s squared, plus 4s plus 20.

00:19 And we're going to carry through the denominator to get that 2s plus 5 has to be equal to a times s squared plus 4s, plus 20 plus b s plus c times s.

00:38 We're going to foil out all the terms together.

00:41 This is a, s squared, plus 4a, s squared, plus 4a, plus 20a, plus b, s squared, plus c, s.

00:51 We're going to gather all the terms by powers of s.

00:54 So this is a plus b times s squared, plus 4a plus c, all times s, s, plus 20 a has to be equal to 2 s plus 5.

01:08 We also have a 0 times s squared here since that will just be 0.

01:13 Now we should have to match coefficients.

01:15 So matching the s squared, a plus b has to equal 0 or a has to be equal negative b.

01:22 For the second one, for a plus c has to be equal to 2.

01:27 And for the non -s term 20 a has to be equal to 5.

01:34 We can still.

01:34 Start with this one.

01:35 So therefore, a is 5 divided by 20, which is 1 fourth.

01:41 Therefore, in this equation, 4 times 1 fourth is 1 plus c has to equal 2, or c is equal to 1.

01:48 And here, b is equal to negative a, which is negative 1 4th.

01:55 So therefore, our fraction will be plugging in 1 4th for a, 1 over 4s, plus negative 1 fourth s plus 1, all divided by s squared plus 4 s plus 20, which we can look into that now.

02:20 So if we look at s plus 2 all squares, we're going to complete the square here.

02:25 S plus 2 all squared is s squared plus 4s plus 4.

02:31 To get to our denominator, we need add 16.

02:33 So s plus 2 squared plus 16 is equal to s squared...

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