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Systems of Linear Differential Equations

Systems of Linear Differential Equations

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Systems of Linear Differential Equations

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  1. CHAPTER 10 Systems of Linear Differential Equations

  2. Contents • 10.1 Preliminary Theory • 10.2 Homogeneous Linear Systems • 10.3 Solution by Diagonalization • 10.4 Nonhomogeneous Linear Systems • 10.5 Matrix Exponential

  3. 10.1 Preliminary Theory • IntroductionRecall that in Sec 3.11, we have the following system of linear DEs: (1)

  4. Consider the system of first-order DEs (2)

  5. Linear Systems • When (2) is linear, we have the normal form as (3)

  6. Matrix Form of a Linear Systems • If we letthen (3) becomes (4)If it is homogeneous, (5)

  7. Example 1 (a) If , the matrix form of is

  8. (b) If , the matrix form of is

  9. DEFINITION 10.1 Solution Vector A solution vector on an interval I is any column matrix whose entries are differentiable functions satisfying (4) on the interval.

  10. Example 2 • Verify that on (−, ) are solutions of (6)

  11. Example 2 (2) SolutionFromwe have

  12. Initial Value Problem (IVP) • Let Then the problem with X(t0) = X0 (7)is an IVP.

  13. THEOREM 10.1 Let the entries of A(t) and F(t) be continuous functions on a common interval I that contains t0, Then there exists a unique solution of (7) on I. Existence of a Unique Solution THEOREM 10.2 Let X1, X2,…, Xk be a set of solutions of the homogeneous system (5) on I, thenX = c1X1 + c2X2 + … + ckXkis also a solution on I. Superposition Principles

  14. Example 3 • Please verify that are solutions of (8)

  15. Example 3 (2) then is also a solution.

  16. DEFINITION 10.2 Let X1, X2, …, Xk be a set of solution vectors of the homogeneous system (5) in an interval I. We say the set is linear dependent in the interval if there exist constants c1, c2, …, ck, not all zero, such that c1X1 + c2X2 + … + ckXk = 0 for every t in the interval. If the set of vectors is not linearly dependent on the interval, it is said to be linearly independent. Linear Dependence/Independence

  17. THEOREM 10.3 Letbe n solution vectors of the homogeneous system (5) on an interval I. Then the set of solution vectors is linearly Independent on I if and only if the Wronskian Criterion for Linearly Independent Solution

  18. (continued) THEOREM 10.3 (9)for every t in the interval. Criterion for Linearly Independent Solution

  19. (continued) If are linearly dependent on I, then W=0 at every point of I. If are linearly independent on I, then W 0 at every point of I. Criterion for Linearly Independent Solution

  20. Example 4 • We saw thatare solutions of (6). Since they are linearly independent for all real t.

  21. DEFINITION 10.3 Any set of n linearly independent solution vectors X1, X2, …, Xnof the homogeneous system (5) on an Interval I is said to be a fundamental set of solutions on the interval. Fundamental Set of Solution

  22. THEOREM 10.4 There exists a fundamental set of solutions for the homogeneous system (5) on an interval I. Existence of a Fundamental Set THEOREM 10.5 Let X1, X2, …, Xn be a fundamental set of solutions of the homogeneous system (5) on an interval I. Then the general solution of the system in the interval is X = c1X1 + c2X2 + … + cnXnwhere the ci, i = 1, 2,…, n are arbitrary constants. General Solution—Homogeneous Systems

  23. Example 5 • We saw thatare linearly independent solutions of (6) on (−, ). Hence they form a fundamental set of solutions. Then general solutions is then (10)

  24. Example 6 • Consider the vectors

  25. Example 6 (2) • Their Wronskian is

  26. Example 6 (2) • Then the general solution is

  27. THEOREM 10.6 General Solution—Nonhomogeneous Systems Let Xp be a given solution of the nonhomogeneous system (4) on the interval I, and let Xc = c1X1 + c2X2 + … + cnXndenote the general solution on the same interval of the associated homogeneous system (5). Then the general solution of the nonhomogeneous system on the interval is X = Xc + Xp. The general solution Xc of the homogeneous system (5) is called the complementary function of the nonhomogeneous system (4).

  28. Example 7 The vector is a particular solution of (11)on (−, ). In Example 5, we saw the solution of isThus the general solution of (11) on (−, ) is

  29. 10.2 Homogeneous Linear Systems • A QuestionWe are asked whether we can always find a solution of the form (1)for the homogeneous linear first-order system (2)

  30. Eigenvalues and Eigenvectors • If (1) is a solution of (2), then X = Ket then (2) becomes Ket = AKet . Thus we have AK = K or AK – K = 0. Since K = IK, we have (A – I)K = 0 (3) The equation (3) is equivalent to

  31. If we want to find a nontrivial solution X, we must have det(A – I) = 0The above discussions are similar to eigenvalues and eigenvectors of matrices.

  32. 8.8 The Eigenvalue Problems DEFINITION 8.13 Let A be n  n matrix. A number  is said to be an eigenvalue of A if there exists a nonzero solution vector K of AK = K(1)The solution vector K is said to be an eigenvector corresponding to the eigenvalue . Eigenvalues and Eigenvectors

  33. Procedure for finding a solution Step 1: Determine the eigenvalue . Step 2: Determine the associated eigenvector by solving AK= K Step 3: Form a solution K .

  34. THEOREM 10.7 Let 1, 2,…, n be n distinct eigenvalues of the matrix A of (2), and let K1, K2,…, Kn be the corresponding eigenvectors. Then the general solution of (2) is General Solution—Homogeneous Systems

  35. THEOREM Let 1, 2,…, n be n distinct eigenvalues of the matrix A of (2), and let K1, K2,…, Kn be the corresponding eigenvectors. Then K1, K2,…, Kn are linearly Independent.

  36. Example 1 Solve (4) Solutionwe have 1 = −1, 2 = 4.

  37. Example 1 (2) For 1 = −1, we have 3k1 + 3k2 = 0 2k1 + 2k2 = 0Thus k1 = – k2. When k2 = –1, then For 1 = 4, we have −2k1 + 3k2 = 0 2k1 − 2k2 = 0Thus k1 = 3k2/2. When k2 = 2, then

  38. Example 1 (3) We have and the solution is (5)

  39. Example 2 Solve (6) Solution

  40. Example 2 (2) • For 1 = −3, we have Thus k1 = k3, k2 = 0. When k3 = 1, then (7)

  41. Example 2 (3) • For 2 = −4, we have Thus k1 = 10k3, k2 = − k3. When k3 = 1, then (8)

  42. Example 2 (4) • For 3 = 5, we have Then (9)

  43. Example 2 (5) • Thus

  44. Example 3: Repeated Eigenvalues Solve Solution

  45. Example 3 (2) (repeated eigenvalue is complete) • We have – ( + 1)2(– 5) = 0, then 1 = 2 = – 1,3 = 5.For 1 = – 1,k1 – k2 + k3 = 0 or k1 = k2 – k3. Choosing k2 = 1, k3 = 0 and k2 = 1, k3 = 1, in turn we have k1 = 1 and k1 = 0.

  46. Example 3 (3) • Thus the two linearly independent eigenvectors areFor 3 = 5,

  47. Example 3 (4) • Implies k1 = k3 and k2 = – k3. Choosing k3 = 1, then k1 = 1, k2 = –1, thusThen the general solution is

  48. DEFINITION 7.7 A set of vectors {x1, x2, …, xn} is said to be linearly independent, if the only constants satisfying k1x1 + k2x2 + …+ knxn = 0 (3) are k1=k2 = … = kn = 0. If the set of vectors is not linearly independent, it is linearly dependent. Linear Independence

  49. Repeated eigenvalue is defective • Suppose 1 is of multiplicity 2 and there is only one eigenvector. A second solution will be of the form (12)Thus X = AX becomeswe have (13) (14)

  50. Example 4 Solve SolutionFirst solve det (A – I) = 0 = ( + 3)2,  = -3, -3, and then we get the first solutionLet