Mathe III Lecture 7 Mathe III Lecture 7

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

3

4

5

6 Linear Equations

7 Linear Homogeneous Equation

8

9

10

11

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

13 The solutions:

14

15 Example:

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

17 The nonhomogeneous equation search for a polynomial solution

18

19 The nonhomogeneous equation

20

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

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

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