1. Mathe III Lecture 4 Mathe III Lecture 4 Mathe III Lecture 4 Mathe III Lecture 4

2. 2 There is no lecture on December 3rd There is an additional lecture on Friday 28 th November 16:00 – 19:00 HS C

3. 3 We have shown that: Hence: If we have two independent solutions, then we have ALL.

4. 4 The nonhomogeneous equation

5. 5 Lemma: Proof:

6. 6 We have shown that:

7. 7 Or equivalently:

8. 8 A simple geometric analogue Consider the equation: and the associated homogeneous equation: The set of solutions to the homogeneous equation is a linear space

9. 9 A simple geometric analogue x y y = 5x + = y = 5x +15 1 20

10. 10 Second Order equations with Constant coefficients.

11. 11 Second Order equations with Constant coefficients.

12. 12 Second Order equations with Constant coefficients.

13. 13 Second Order equations with Constant coefficients. {{ 0 0

14. 14 Second Order equations with Constant coefficients.

15. 15 Second Order equations with Constant coefficients. This can be worked out directly by looking at the complex solutions m 1 t, m 2 t or by using trigonometry to show that the two expressions above are solutions and are independent

16. 16 Second Order equations with Constant coefficients. Summary: The general solution of a second order equation is: (for arbitrary A,B)

17. 17 Second Order equations with Constant coefficients. Nonhomogeneous equations We need to find one particular solution The general solution will be: Consider the case c t is a constant Search, first, for a constant solution:

18. 18 Second Order equations with Constant coefficients. Nonhomogeneous equations Consider the case c t is a constant What if Search for a linear solution:

19. 19 Second Order equations with Constant coefficients. Nonhomogeneous equations Consider the case c t is a constant What if Search for a quadratic solution:

20. 20 Consider the case c t is a linear combination of terms of the form a t, t m Undetermined Coefficients Example: Look for a solution of the form:

21. 21

22. 22 is a particular solution

23. 23 A general solution to the homogeneous equation Solve: A general solution of the equation is:

24. 24 Stability: In the long run, the solution should be independent of the initial conditions. The general solution of is: if : The system is stable.

25. 25 if 1 m