Matrix 1st Order Linear Ordinary Differential Equation Problem Solution

Here we solve a 1st Order Linear Ordinary Differential Equation with 2x2 Matrix or 2x1 Vector Coefficients. The independent variable is t and the dependent variable is X.

The procedure is shown here in the chapters.

0:00 Start
0:28 1st Order Linear ODE Statement
2:09 Definitions
3:12 Method of Undetermined Coefficients
4:13 Homogeneous Solution
4:51 General Formulas for Solving
6:24 Homogeneous Solution Formula
7:05 Matrix Exponential
9:21 Diagonalizing Matrix
10:28 Particular Solution Formula
11:38 Unique Eigenvectors Required
12:30 Particular Solution must not match Homogenous
13:28 Solve: Find eigenvalues of A
15:45 Theory of eigenvectors
17:03 Solve: Find eigenvectors of A
21:19 Solve: Find diagonalizing matrix P
21:50 Solve: Find inverse of matrix P
23:29 Solve: Calculate matrix exponential
25:17 Solve: Find homogenous solution.
26:45 Start of particular solution
26:52 Solve: Particular solution, first guess
28:47 Solve: Particular solution, second guess
31:45 Solve: Second guess derivative
32:33 Solve: Insert derivative and second guess to main equation
33:33 Solve: Split into 3 equations
33:16 Solve: t^2 equation
36:47 Solve: t equation and constants equation
38:30 Discussion of non-singular matrices
40:30 Combine homogeneous and particular solutions
42:59 End credits

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