1. Lecture 3: Solving Diff Eqs with the Laplace Transform
Solving differential equations with the Laplace transform
Inverse Laplace transform
Partial fraction expansion
Case I: Poles distinct
Case II: Repeated poles
Case III: Complex poles
MATLAB introduction
ME 431

2. Solving LTI Differential Equations
The Laplace transform converts Linear Time-Invariant (LTI) differential equations to algebraic
solve in time domain
differential
equation
x(t) L
L-1
ME 431
algebraic
equation
solve in s-domain
X(s)

3. Inverse Laplace Transform
Item No.
f(t)
F(s)
δ(t)
1(t) t tn
e-at
sin (ωt)
cos (ωt) 1. 2. 3. 4. 5. 6. 7.
The inverse Laplace transform can be solved for using the table and properties introduced earlier
ME 431

4. Example δ(t) 1(t) t tn e-at sin (ωt) cos (ωt) 1. 2. 3. 4. 5. 6. 7.
Item No.
f(t)
F(s)
δ(t)
1(t) t tn
e-at
sin (ωt)
cos (ωt) 1. 2. 3. 4. 5. 6. 7.

5. Example

6. Inverse Laplace Transform
Laplace domain expressions are generally a ratio of two polynomials
Want to rearrange into a form that is recognizable in the table
 partial fraction expansion
ME 431

7. Partial Fraction Expansion
Split rational functions into simpler terms (reverse of finding a common denominator)
Need to consider three different cases
ME 431
If by hand use first form,
If by MATLAB use second

8. Partial Fraction Expansion
Let m be the degree of the numerator and n be the degree of the denominator
If (m < n) can begin
If (m ≥ n) do long division first
Begin by finding the roots of the denominator (poles), determines case to employ
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9. Case I: Distinct Poles
Example:

10. Case I: Distinct Poles (cont’d)

11. Case II: Repeated Poles
Example:

12. Case II: Repeated Poles (cont’d)

13. Case III: Complex Poles
Example:
Can solve like Case I
Or can solve as follows

14. Case III: Complex Poles (cont’d)

15. Case III: Complex Poles (cont’d)

16. Solving LTI Differential Equations
Recall the solution procedure
solve in time domain
differential
equation
x(t) 1 L
L-1 3
ME 431
algebraic
equation
solve in s-domain
X(s) 2

17. Example
Solve

18. Example (continued)

19. MATLAB Commands
ME 431
Other MATLAB basics, vectors, matrices, multiplication, division, powers, functions, element wise, plotting