1. Math 4B Systems of Differential Equations Matrix Solutions Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. A system of 1 st order linear differential equations will have the form Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Here we are dealing with n-vectors and an nxn matrix A.

3. The nxn system is equivalent to a single nth-order differential equation. The x i functions are simply derivatives of the x i-1 functions. Here is an example: Convert the following equation into a system of 1 st -order equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

4. The nxn system is equivalent to a single nth-order differential equation. The x i functions are simply derivatives of the x i-1 functions. Here is an example: Convert the following equation into a system of 1 st -order equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Define variables as follows:

5. The nxn system is equivalent to a single nth-order differential equation. The x i functions are simply derivatives of the x i-1 functions. Here is an example: Convert the following equation into a system of 1 st -order equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Define variables as follows: The original equation becomes: Now we can write down the vector equation.

6. The nxn system is equivalent to a single nth-order differential equation. The x i functions are simply derivatives of the x i-1 functions. Here is an example: Convert the following equation into a system of 1 st -order equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Define variables as follows: The original equation becomes: Now we can write down the vector equation.

7. To find the solution to this system, find the eigenvalues and eigenvectors. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Recall that an eigenvalue is a constant λ such that For some corresponding eigenvector. This equation will be true if the determinant of (A-λI) is 0. This is the characteristic equation.

8. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Find the roots of the characteristic equation to find the eigenvalues. After a bit of trial and error, notice that λ=-1 is an eigenvalue. This means that (λ+1) is a factor. Divide it out to get the remainder (a quadratic). Next find an eigenvector for each eigenvalue. Plug back into the matrix and row reduce:

9. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB For λ=-1: Any multiple of this will also be an eigenvector for λ=-1.

10. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB For λ=-4: Any multiple of this will also be an eigenvector for λ=-4.

11. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB For λ=3: Any multiple of this will also be an eigenvector for λ=3.

12. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Here are the eigenvalues and their associated eigenvectors: The solution to the system is:

13. Example: Find the solution to the given initial value problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

14. Example: Find the solution to the given initial value problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Converting to matrix form, we have:

15. Example: Find the solution to the given initial value problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Converting to matrix form, we have: We need the eigenvalues and eigenvectors for the matrix:

16. Example: Find the solution to the given initial value problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Converting to matrix form, we have: We need the eigenvalues and eigenvectors for the matrix: Eigenvector for λ=3 solves: Any multiple of this will work as an eigenvector

17. Example: Find the solution to the given initial value problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Converting to matrix form, we have: We need the eigenvalues and eigenvectors for the matrix: Eigenvector for λ=4 solves: Any multiple of this will work as an eigenvector

18. Example: Find the solution to the given initial value problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Converting to matrix form, we have: The general solution is: Use the initial values to find c 1 and c 2.

19. Example: Find the solution to the given initial value problem. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Converting to matrix form, we have: The general solution is: Use the initial values to find c 1 and c 2.

20. Example: Find the solution to the given system. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

21. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Next find eigenvectors for the eigenvalues. Example: Find the solution to the given system.

22. For λ=-1+4i Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB It will help to split the vector up into real and imaginary parts, as shown.

23. For λ=-1+4i Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB It will help to split the vector up into real and imaginary parts, as shown. The e-vector for the other e-value will be the complex conjugate. For λ=-1-4i

24. Now there will be a solution corresponding to each eigenvalue: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Using Euler’s Formula these can be written as sines and cosines: Get real solutions by combining these:

25. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB The general solution is a linear combination of the real-valued solutions: The solution can be packaged more neatly through the use of the “Fundamental Matrix” of independent solutions. Each column of this matrix is one of the independent solutions X (1) and X (2). This is the fundamental matrix.

26. Example: Find the solution to the given system. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB This is a graph of several solution curves. We call this a stable spiral.

27. Example: Find the general solution to the given system of differential equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

28. Example: Find the general solution to the given system of differential equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

29. Example: Find the general solution to the given system of differential equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

30. Example: Find the general solution to the given system of differential equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB