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Definition: For a function f(t) (f(t)=0 for t<0),
Definition: We can transform an ordinary differential
equation (ODE) into an algebraic equation (AE).
t-domain (s: complex variable) s-domain AE ODE
1 f(t) 2
0 t F(s)
Solution to ODE We denote Laplace transform of f(t) by F(s).
3 Partial fraction
4 Properties of Laplace transform
Differentiation (review) Example 1
ODE with initial conditions (ICs) t-domain 1. Laplace transform s-domain
5 Example 1 (cont’d)
2. Partial fraction expansion 6 Example 1 (cont’d) unknowns 3. Inverse Laplace transform Multiply both sides by s & let s go to zero: Similarly, If we are interested in only the final value of y(t), apply
Final Value Theorem: 7 8 Example 2
S1 In this way, we can find a rather
complicated solution to ODEs easily by
using Laplace transform table...
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- Spring '08