MAT 3237 Differential Equations Section 4.4 Additional Operational Properties Part I

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Presentation transcript:

MAT 3237 Differential Equations Section 4.4 Additional Operational Properties Part I

Commercial Math Minor 1. Calculus 1 – 3 2. Additional 15 upper division credits (a) DVQ – 3 (b) Vector Cal – 3

Possible Electives Teaching Minor (MAT 2720 Discrete Math – 3) MAT 3749 Intro to Analysis – 3 MAT 4402 Modern Algebra – 3 MAT 3441 Axiomatic Geometry – 3

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Theorem 4.8

Theoretical Background Laplace transforms do not exist for all functions If f(t) satisfy certain “nice conditions”, its Laplace transform exists (Theorem 4.2) Under the same “nice conditions”, interchanging of differentiation and integration is possible (Advanced Calculus/ Analysis)

Theoretical Background Laplace transforms do not exist for all functions If f(t) satisfy certain “nice conditions”, its Laplace transform exists (Theorem 4.2) Under the same “nice conditions”, interchanging of differentiation and integration is possible (Advanced Calculus/ Analysis)

Theoretical Background Laplace transforms do exist for all functions If f(t) satisfy certain “nice conditions”, its Laplace transform exists (Theorem 4.2) Under the same “nice conditions”, interchanging of differentiation and integration is possible (Advanced Calculus/ Analysis)

Theorem 4.8

Proof:

Example 1

Before Example 2….. Rewrite our formula for more complicated situations

Example 2

Next Question… Given functions F(s) and G(s). What is the function h(t) such that

Definition: Convolution

Convolution This type of integrals has many applications in physics and engineering Signal and System Analysis

Convolution

Combining the 2 facts: 2 Forms of Convolution Theorem For Part II……

Example 3

Warning