1 Section 2 SECTION 2 Partial Fractions

2 We need to split the following into separate terms: Roots of the denominator D(s): Case I – unrepeated factor Case II – repeated factor Case III – complex factors Case IV – repeated complex factors Section 2

3 Case I – unrepeated factor example Section 2

4 Use the "cover up" rule …

5 Section 2 Inverse Laplace Transform:

6 Section 2 example Inverse Laplace Transform:

7 Section 2 Why does the "cover up" rule work ? alternative method solve these

8 Question: Section 2 Obtain the Partial Fractions to

9 Section 2 Case II – repeated factor example unrepeated factor repeated factor

10 Section 2 Use the repeated factor rule …

11 Section 2 Inverse Laplace Transform:

12 Section 2 Case III – complex factors example complex conjugates Inverse Laplace Transform:

13 Section 2 example Inverse Laplace Transform: where

14 Section 2 (1)Attacking Polynomials Directly (rather than using the complex formulae) Other Topics not Covered Inverse Laplace Transform: using

15 Section 2 (2) To solve these, you just use the methods of cases II and III, only with complex number calculations Case IV – repeated complex factors