# Math 0250 Homework 6

• 2.3 Number 10
The eigenvalues are 1, 2i, and -2i, with corresponding eigenvectors

The eigenvalue 1 gives the solution

The eigenvalue 2i gives the complex solution

Taking real and imaginary parts gives the real solutions

so the general solution to the differential equation is

with initial value

Thus, to get a solution satisfying any of the given initial conditions, just replace the constants c1, c2, and c3 by the coordinates of the initial vector.

• 2.4 Number 1
The eigenvalues of A are 2 and -2, with corresponding eigenvectors

respectively, so a matrix solution is

Putting t=0 gives

so

and so a fundamental matrix is

• 2.4 Number 9
The eigenvalues of A are 2 and 1, with corresponding eigenvectors

respectively, so a matrix solution is

Putting t=0 gives

so

and so a fundamental matrix is

• 2.4 Number 14
The eigenvalues of A are 1, 2, and 0, with corresponding eigenvectors

respectively, so a matrix solution is

Putting t=0 gives

so

and so a fundamental matrix is

• 2.4 Number 17
Using the fundamental matrix found in problem 1:

• 2.4 Number 19
Using the fundamental matrix found in problem 1:

• 2.5 Number 6
Let . Then

Frank Beatrous
1998-10-25