1. Mathe III Lecture 7 Mathe III Lecture 7

2. 2 Second Order Differential Equations The simplest possible equation of this type is:

3. 3

4. 4

5. 5

6. 6 Linear Equations

7. 7 Linear Homogeneous Equation

8. 8

9. 9

10. 10

11. 11

12. 12 Linear Equations with constant coefficients The homogeneous equation: search a solution of the form:

13. 13 The solutions:

14. 14

15. 15 Example:

16. 16 The nonhomogeneous equation We need to find a particular solution search for a constant solution

17. 17 The nonhomogeneous equation search for a polynomial solution

18. 18

19. 19 The nonhomogeneous equation

20. 20

21. 21 Stability: The general solution to: is Globally Asymptotically Stable (Stable) if:

22. 22 For second order linear equations with constant coefficients: two independent solutions for the homogeneous equation are:

23. 23 Constrained Maximization Lagrange Multipliers At a maximum point of the original problem the derivatives of the Lagrangian vanish (w.r.t. all variables).