1. Find the general solution of the differential equation

2. Solve the system

with the initial condition

3. For the matrix

find the eigenvalues and their algebraic and geometric multiplicities. Identify the eigenvectors and, if applicable, generalized eigenvectors.

4. The system

is solved using the Euler method with step size . Find the resulting approximation for .

Answers:

1. .

2. , .

3. The eigenvalues are 2 (algebraically double, geometrically simple) and 3 (simple). The eigenvector for 3 is (3,2,1), and the eigenvector for 2 is (1,0,0). A generalized eigenvector is given by (0,1,0).

4. .

Michael Renardy 2004-09-21