Sec. 7.4: Basic Theory of Systems of First Order Linear Equations MATH 351 California State University, Northridge April 2, 214 MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 1 / 12
System of First Order Linear Equations A system of n first order linear equations: x 1 = p 11(t)x 1 + + p 1n(t)x n + g 1(t) x 2 = p 21(t)x 1 + + p 2n(t)x n + g 2(t). x n = p n1(t)x 1 + + p nn(t)x n + g n(t) To discuss the system (1) most effectively, we write it in matrix notation. That is, x 1(t) g 1(t) p 11(t) p 12(t) p 1n(t) x 2(t) x(t) =., g(t) = g 2(t)., P(t) = p 21(t) p 22(t) p 2n(t)... x n(t) g n(t) p n1(t) p n2(t) p nn(t) Then the system (1) becomes (1) x = P(t)x + g(t) (2) A vector x(t) = φ(t) is said to be a solution of Eq. (2) if its components satisfy the system of equations (1), that is x 1 = φ 1(t), x 2 = φ 2(t),..., x n = φ n(t), (3) which can be viewed as a set of parametric equations in an n-dim space. MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 2 / 12
Recall the Existence and Uniqueness Theorem in Sec. 7.1 Theorem 7.1.2 If the functions p 11, p 12,..., p nn, g 1,..., g n are continuous on an open interval I : α < t < β, then there exists a unique solution x 1 = φ 1(t),..., x n = φ n(t) of the system (1) that also satisfies the initial conditions x 1(t ) = x 1, x 2(t ) = x 2,, x n(t ) = x n, (4) where t is any point in I, and x 1, x 2,, x n are any prescribed numbers. Moreover, the solution exists throughout the interval I. Throughout this section, we assume that P and g are continuous on some interval α < t < β; that is, each of the scalar functions p 11, p 12,..., p nn, g 1,..., g n is continuous on α < t < β. MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 3 / 12
Homogeneous Eqn and Nonhomogeneous Eqn If g(t) =, then x = P(t)x is homogeneous. If g(t), then x = P(t)x + g(t) in nonhomogeneous. (Section 7.9) We use the notation x 11(t) x 1k (t) x (1) x 21(t) (t) =.,, x 2k (t) x(k) (t) =., (5) x n1(t) x nk (t) to designate specific solutions of the homogeneous system. NOTE: x ij (t) = x (j) i (t) refers to the ith component of the jth solution x (j) (t). The structure of solutions of system are stated in the following 5 theorems. x = P(t)x (6) MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 4 / 12
Principle of Superposition Theorem 7.4.1 If the vector functions x (1) and x (2) are solutions of the system (6), then the linear combination c 1x (1) + c 2x (2) is also a solution for any constants c 1 and c 2. NOTE: This can be proved simply by differentiating c 1x (1) and c 2x (2) and using the fact that x (1) and x (2) satisfy Eq. (6). REMARK: By repeated application of Theorem 7.4.1, we can conclude that if x (1),..., x (k) are solutions of Eq. (3), then x = c 1x (1) (t) + + c k x (k) (t) is also a solution for any constants c 1,..., c k. QUESTION: whether all solutions of Eq. (6) can be found in this way. MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 5 / 12
General Solutions If we properly choose n solutions for the system (6) of n 1st-order eqns, will the linear combinations of these solutions be sufficient? Let x (1),..., x (n) be n solutions of the system (6), and consider the matrix x 11(t) x 1n(t) X(t) =.. (7) x n1(t) x nn(t) Recall from Sec. 7.3 that the columns of X(t) are linearly independent for a given value of t iff det X for that value of t. This determinant is called the Wronskian of the n solutions x (1),..., x (n) (denoted by W [x (1),..., x (n) ]), that is W [x (1),..., x (n) ] = det X(t) (8) Therefore, the solutions x (1),..., x (n) are linearly independent at a point iff W [x (1),..., x (n) ] is not zero there. MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 6 / 12
General Solution and Fundamental Set of Solutions Theorem 7.4.2 If the vector functions x (1),..., x (n) are linearly independent solutions of the system (6) for each point in the interval α < t < β, then each solution x = φ(t) of the system (6) can be expressed as φ(t) = c 1x (1) (t) + + c nx (n) (t) (9) in exactly one way. general solution: c 1x (1) (t) + + c nx (n) (t). fundamental set of solutions: any set of solutions of x (1),..., x (n) of is linearly independent Eq. (6) at each point in the interval α < t < β. MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 7 / 12
Proof: We will show that any solution φ(t) of Eq. (6) can be written as for suitable values of c 1,..., c n. φ(t) = c 1x (1) (t) + + c nx (n) (t) MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 8 / 12
Theorem 7.4.3 If x (1),..., x (n) are sols of Eq. (6) on the interval α < t < β, then in this interval W [x (1),..., x (n) ] either in identically zero or else never vanishes. Proof: We need to first show that (see Problem 2) dw dt = [p 11(t) + p 22(t) + + p nn(t)]w MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 9 / 12
The existence of fundamental set of solutions Theorem 7.4.4 Let 1 1 e (1) =, e (2) =,, e (3) =... 1 further, let x (1),..., x (n) be the solutions of the system (6) that satisfy the initial conditions x (1) (t ) = e (1),..., x (n) (t ) = e (n), (1) respectively, where t is any point in α < t < β. Then x (1),..., x (n) form a fundamental set of solutions of the system (6). Proof: the existence and uniqueness of the solutions x (1),..., x (n) are ensured by Theorem 7.2.1 W [x (1),..., x (n) ] = MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 1 / 12
Theorem 7.4.5 Consider the system (6) x = P(t)x where each element of P is a real-valued continuous function. If x = u(t) + iv(t) is a complex-valued solution of Eq. (6), then its real part u(t) and its imaginary part v(t) are also solutions of this equation. Proof: MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 11 / 12
Summary 1 Any set of n linearly independent solutions of the system x = P(t)x constitutes a fundamental set of solution. 2 Under the conditions given in this section, such fundamental sets always exists. 3 Every solution of the system x = P(t)x can be represented as a linear combination of any fundamental set of solutions. MATH 351 (Differential Equations) Sec. 7.4 April 2, 214 12 / 12