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

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)

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

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.

Example

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

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

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 ME 431

Case I: Distinct Poles Example:

Case I: Distinct Poles (cont’d)

Case II: Repeated Poles Example:

Case II: Repeated Poles (cont’d)

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

Case III: Complex Poles (cont’d)

Case III: Complex Poles (cont’d)

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

Example Solve

Example (continued)

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