In Problems 1 and 2, fill in the blanks.

1. The column vector X = k is a solution of the linear system X′ = for k = .
2. The vector X = c₁ e^−9t + c₂ e^7t is a solution of the initial-value problem X′ = X, X(0) = for c₁ = and c₂ = .
3. Consider the linear system X′ = X. Without attempting to solve the system, which one of the following vectors,

   

   is an eigenvector of the coefficient matrix? What is the solution of the system corresponding to this eigenvector?

4. Consider the linear system X′ = AX of two differential equations where A is a real coefficient matrix. What is the general solution of the system if it is known that λ₁ = 1 + 2i is an eigenvalue and K₁ = is a corresponding eigenvector?

In Problems 5–14, solve the given linear system by the methods of this chapter.

5. = 2x + y
   = −x
6. = −4x + 2y
   = 2x − 4y
7. X′ = X
8. X′ = X
9. X′ = X
10. X′ = X
11. X′ = X +
12. X′ = X +
13. X′ = X +
14. X′ = X + e^2t
15. 1. Consider the linear system X′ = AX of three first-order differential equations where the coefficient matrix is

       

       and λ = 2 is known to be an eigenvalue of multiplicity two. Find two different solutions of the system corresponding to this eigenvalue without using any special formula (such as (12) of Section 10.2).

    2. Use the procedure in part (a) to solve

       X′ = X.

16. Verify that X = e^t is a solution of the linear system

    X′ = X

    for arbitrary constants c₁ and c₂. By hand, draw a phase portrait of the system.