University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2021
Views:

106

Pages:

1
Author:

Reece Kane

Math 244 Homework To Replace 7.2 1. Exercise 22 (Section 7.1, p 363) 2. Give the solution to each system. If it has an infinite number of solutions, give your answer in vector form: 3x + 2y = 1 3x + 2y = 1 3x + 2y = 1 2x − y = 3 6x + 4y = 3 6x + 4y = 2 3. Write each of the previous systems in matrix-vector form. Verify that the determinant of the first matrix is not zero, but is zero for the second and third. 4. Write each system of differential equations in matrix-vector form or write the system from the matrixvector form: x01 = 3x1 − x2 −1 2 0 x = x x02 = 9x1 − 3x2 2 1 5. Find the equilibrium solutions to the previous autonomous linear differential equations. For the following problems, define 1 A= −1 2 1 b= −1 1 B= 3 1 1 3 6. Similar to Exercise 1 (Section 7.2). Given the matrices A and B above: 7. Compute A − 4B 11. Compute A−1 b 8. Compute (B − 4I)b 12. Verify: Ab − 3b = (A − 3I)b 9. Compute det(B − 4I) 13. Compute B T B, Tr(A), and Tr(B) 14. Is AT = A? Is B T = B? 10. Are AB and BA the same? 15. Solve the system of complex equations. Before we start, check that we get the bottom equation if we divide the top equation by (1 + i). −(1 + i)v1 −v1 +2v2 +(1 − i)v2 =0 =0 16. Solve the system of complex equations. Before we start, check that we get the bottom equation if we divide the top equation by (2 − i). (2 − i)v1 v1 −5v2 −(2 + i)v2 =0 =0 17. Exercises 22, 23 (Section 7.2, p. 373) 3 −2 4 0 (a) If x = x, verify that a solution is x(t) = e2t 2 −2 2 1 2 2 −1 1 et + te2t (b) If x0 = x+ et , verify that a solution is x(t) = 3 −2 −1 0 2 1

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