1. MAT 3237 Differential Equations Section 4.4 Additional Operational Properties Part I http://myhome.spu.edu/lauw

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

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

4. Possible Electives (Applied Math) Minor – Analysis not required MAT 3360 Probability & Statistics – 5 MAT 4830 Mathematical Modeling – 5 MAT 4725 Numerical Analysis – 5 Possible petition as Technical Electives Credits

5. Review & Preview

7. Theorem 4.8

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)

9. 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)

10. 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)

11. Theorem 4.8

12. Proof:

13. Example 1

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

16. Example 2

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

20. Definition: Convolution

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

22. Convolution

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

25. Example 3

27. Warning