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Chapter 7: Problem 13

Express in partial fractions $$ G(s)=\frac{6(s+2)}{(s+1)(s+3)} $$

Short Answer

Expert verified
Answer: The partial fraction decomposition of the given function G(s) is G(s) = 3/(s + 1) + 3/(s + 3).

Step by step solution

01

Identify the partial fractions form

The given function is: $$ G(s) = \frac{6(s + 2)}{(s + 1)(s + 3)} $$ We express G(s) as the sum of two simpler fractions with linear denominators: $$ G(s) = \frac{A}{s + 1} + \frac{B}{s + 3} $$ Where A and B are constants to be determined.
02

Clear the denominators

We multiply both sides of the equation by the original denominator, \((s + 1)(s + 3)\), to eliminate the denominators and obtain a linear equation in s: $$ 6(s + 2) = A(s + 3) + B(s + 1) $$
03

Solve for the constants A and B

There are multiple ways to find the values for A and B. We will use the method of substitution here, by choosing appropriate values for s: 1) Let s = -1: $$ 6(-1 + 2) = A(-1 + 3) + B(-1 + 1) \Rightarrow 6 = 2A \Rightarrow A = 3 $$ 2) Let s = -3: $$ 6(-3 + 2) = A(-3 + 1) + B(-3 + 3) \Rightarrow -6 = -2B \Rightarrow B = 3 $$ Now, we have our values for A and B: A = 3 and B = 3.
04

Write the partial fractions decomposition

Now that we have found the values of the constants A and B, we substitute them back into our expression for G(s): $$ G(s) = \frac{3}{s + 1} + \frac{3}{s + 3} $$ This is the partial fraction decomposition of the given function G(s).

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Most popular questions from this chapter

Rewrite each of the statements without using a modulus sign: $$ |x| \leq 7.5 $$

For each fraction state the degrees of the numerator and denominator, and hence determine which are proper and which are improper: (a) \(\frac{x+1}{x}\) (b) \(\frac{x^{2}}{x^{3}-x}\) (c) \(\frac{(x-1)(x-2)(x-3)}{x-5}\)

Express \(\frac{x^{2}+2 x-1}{x^{3}-x}\) in partial fractions.

Factorise \(t^{3}+3 t^{2}+2 t\).

Calculate the poles of the rational function $$ G(s)=\frac{s+5}{s^{2}+3 s+2} $$
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