1. MAE 82 – Engineering Mathematics
Class Notes 17: System of First Order Linear Differential Equations - Nonhomogenous System(2/2)
MAE 82 – Engineering Mathematics

2. Nonhomogeneous Linear Systems
General Solution
Particular Solution – Methods
Undetermined Coefficients (Quick)
Variation of Parameters (More powerful)
Complementary solution or General solution of the associated homogeneous linear system X’=AX
Particular solution of the nonhomogeneous linear system X’=AX+F

3. Undetermined Coefficients
Assumptions
Constants
Polynomials
Exponential functions
sin/cos
finite sum/products of these functions
Constants

4. Undetermined Coefficients – Example (Constants)
Example (Xp - constants)
on (-∞, ∞)
Solve the associated homogeneous system
(*)

5. Undetermined Coefficients – Example (Constants)
from (*)
Since F(t) is a constant vector, we assume a constant particular solution

6. Undetermined Coefficients – Example (Constants)
Plug into

7. Undetermined Coefficients – Example (First Order Polynomial)
Example (Xp - polynomial)
on (-∞, ∞)
The associated homogeneous system
since the particular solution of
the same form is

8. Undetermined Coefficients – Example (First Order Polynomial)

9. Undetermined Coefficients – Example (Exponential Function)
Example (Xp – exponential functions)
Assumption for a particular solution would be

10. Variation of Parameters
Fundamental Matrix If
X1, X2,… Xn is a fundamental set of solution of the homogeneous system X’=AX on the interval I
Then
Its general solution on the interval is
The general solution can be written as a product

11. Variations of Parameters – Fundamental Matrix
n x 1 column vector of arbitrary constants
n x n matrix, whose columns consist of the entries of the
solution vectors of the system x' = Ax
is called a fundamental matrix

12. Variations of Parameters – Fundamental Matrix
Properties of the fundamental matrix
A fundamental matrix is nonsingular.
If is a fundamental matrix of the system x' = Ax
then,
Variation of parameters
Is it possible to replace the vector of constants C in
by a vector of functions.

13. Variations of Parameters
is a particular solution of the nonhomogeneous system.
by the product rule the derivative of
[Note that the order is important

14. Variations of Parameters
substitute
use the property that

15. Variations of Parameters
Since

16. Example - Variations of Parameters
The associated homogeneous system

17. Example - Variations of Parameters

18. Example - Variations of Parameters

19. Initial Value Problem
Because the limits of integration are chosen so that the particular solution vanishes , substituting into (*) yields
for which we can get