1. MAT 3237 Differential Equations
Section 4.4
Additional Operational Properties Part I

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

3. Review & Preview

4. Review & Preview

5. Theorem 4.8

6. 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)

7. 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)

8. 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)

9. Theorem 4.8

10. Proof:

11. Example 1

12. Example 1

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

14. Example 2

15. Example 2

16. Example 2

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

18. Definition: Convolution

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

20. Convolution

21. Convolution

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

23. Example 3

24. Example 3

25. Warning