Share
Explore BrainMass

Initial-value problem

Using the method of undetermined coefficients, find the solution of the system:

X'=AX + B

that satisfies the initial condition:

X(0)=( 0
1
-1).
A and B are matrices defined in the attached Notepad file.
Note: When solving the homogeneous soln, exhibit a fundamental matrix psi(t) and also the special fundamental matrix phi(t) satisfying phi(0)=I.

Solution Preview

Solution:For this system X'=AX+B (1)
We first need to know the given matrices A and B. If they are matrices with constant numbers, it is easy to solve them. For example, if we know A=P'CP where C=diag(a1,a2,...,an) and P'=P^(-1). P^(-1) is the inverse of P. Note that such P is called orthogonal matrix. Many matrices can easily find such matrix P such that A=P'CP.
Then substituting A=P^(-1)CP , (1) becomes
PX'=CPX+PB (2)
where C=diag{a1,a2,...,an}.
Let Y=PX and D=PB,then (2) is as follows
Y'=CY+D (3)
Obviously Y=-C^(-1)D is a special solution for (3). In order to solve (3), we only need to solve the linear diff. equations (4) as follows
Y'=CY (4)
In fact, IF Y=(y1,y2,...,yn), then (4) can rewrite as follows.
...

Solution Summary

This shows how to find the solution of a system using the method of undetermined coefficients.

$2.19