1. Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs

2. Important Linear Algebra Recall Eigenvalues and Eigenvectors And Linear Independence and Are linearly independent if det

3. Linear Systems of Ordinary Differential Equations Let’s rewrite this in matrix form: Or

4. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? Then :

5. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? Then :

6. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? Then : Or:

7. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? Then : Or:

8. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? Then : Or:

9. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? But these are two independent, separable equations!

10. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? Solution s

11. Linear Systems of Ordinary Differential Equations What ifwas an eigenvector of ? Solutio n But if is an Eigenvector

12. Two Specific Solutions For a 2x2 System with Eigenvalues and Eigenvectors and Specific Solutions: or

13. Example Find Two Solutions to The Set of Linear Differential Equations

14. Linear Combinations of Solutions Remember, For Linear Equations, if is a solution and is a solution, then is also a solution.

15. Linear Combinations of Solutions How can you tell? Remember and

16. Linear Combinations of Solutions How can you tell? Remember and so

17. Linear Combinations of Solutions How can you tell?

18. Linear Combinations of Solutions How can you tell? Because Are Scalars

19. Linear Combinations of Solutions How can you tell? Because Are Scalars

20. Linear Combinations of Solutions How can you tell? SoSo

21. Fundamental Set of Solutions Additionally, if the Eigenvectors are linearly independent Then Form a Fundamental Set of Solutions, is the general solution. det and

22. Why? Consider the Wronskian det Never 0 Only 0 if are linearly dependent det

23. So To Solve Determine All Eigenvalues and Eigenvectors of General Solution takes the form Plug in 0 and use initial conditions to find

24. Summary Eigenvalues and Eigenvectors Can Be Used To Find General Solution of Systems of Equations If Eigenvectors are linearly independent, then we can find general solutions.

25. Questions?