Lecture 31. Basic Theory of First Order Linear Systems

Transcription

1 Math Mathematics of Physics and Engineering I Lecture 31. Basic Theory of First Order Linear Systems April 4, 2012 Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

2 Agenda Existence and Uniqueness Properties and Structure of Solutions of Homogeneous Systems Principle of Superposition Linear Independence of Solutions Wronskian Existence of a Fundamental Set of Solutions Linear n th order ODEs Homework Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

3 Existence and Uniqueness The general first order linear system of dimension n has the following form: where P(t) is an n n matrix and g(t) a n 1 vector. Theorem x = P(t)x + g(t) (1) If P(t) and g(t) are continuous on an open interval I = (α, β), then there exists a unique solution x = z(t) of the initial value problem x = P(t)x + g(t), x(t 0 ) = x 0, (2) where t 0 I, and x 0 is any constant vector with n components. Moreover, the solution exists throughout the interval I. Important Special Case: Consider the following IVP: x = Ax, x(t 0 ) = x 0, (3) where A is a constant n n matrix. The above theorem guarantees that a solution exists and is unique on the entire t-axis. Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

4 Principle of Superposition Consider the homogeneous system Definition x = P(t)x (4) If x 1,..., x k are solutions of system (4), then an expression of the form c 1 x c k x k, (5) where c 1,..., c k are arbitrary constants, is called a linear combination of solutions. Principle of Superposition If x 1,..., x k are solutions of system (4) on the interval I, then any linear combination c 1 x c k x k is also a solution of (4) on I. Using the Principle of Superposition, we can enlarge a finite set of solutions {x 1,..., x k } to a k-dimensional infinite family of solutions c 1 x c k x k parameterized by c 1,..., c k. Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

5 Linear (In)Dependence Goal: to show that all solutions of n-dimensional system x = P(t)x are contained in an n-parameter family c 1 x c n x n, provided that the n solutions x 1,..., x n are distinct (=linearly independent). Definition The n vector functions x 1 (t),..., x n (t) are said to be linearly independent on an interval I if the only constants c 1,..., c n such that for all t I are c 1 = c 2 =... = c n = 0. c 1 x 1 (t) c n x n (t) = 0 (6) If there exist constants c 1,..., c n, not all zero, such that (6) is true for all t I, the vector functions are said to be linearly dependent on I. Example: Show that the following vector functions are linearly independent on I = (, ) e 2t e t x 1 (t) = 0 x 2 (t) = e t e 2t e t Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

6 Wronskian Let x 1,..., x n be n solutions of x = P(t)x and let X(t) be n n matrix whose j th column is x j (t), j = 1,..., n. Definition The Wronskian W = W [x 1,..., x n ] of the n solutions x 1,..., x n is defined by W [x 1,..., x n ](t) = det X(t) (7) Theorem Let x 1,..., x n be solutions of x = P(t)x on an interval I = (α, β) in which P(t) is continuous. If x 1,..., x n are linearly independent on I, then W [x 1,..., x n ](t) 0 at every point in I If x 1,..., x n are linearly dependent on I, then W [x 1,..., x n ](t) = 0 at every point in I Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

7 Structure of Solutions The following theorem shows that all solutions of an n-dimensional homogeneous system are contained in the n-parameter infinite family of solutions, provided that these solutions are linearly independent. Theorem Let x 1,..., x n be solutions of x = P(t)x (8) on an interval I = (α, β) such that, for some point t 0 I, the Wronskian is nonzero, W [x 1,..., x n ](t 0 ) 0. Then each solution x = z(t) of (8) can be written as a linear combination of x 1,..., x n, z(t) = ĉ 1 x 1 (t) ĉ n x n (t), (9) where the constants ĉ 1,..., ĉ n are uniquely defined. Remark: If W [x 1,..., x n ](t 0 ) 0, then c 1 x 1 (t) c n x n (t) is called the general solution. {x 1 (t),..., x n (t)} is called a fundamental set of solutions. Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

8 Existence of a Fundamental Set of Solutions Example: Let A = show that the following solutions of x = Ax form a fundamental set x 1 (t) = e 2t 0 x 2 (t) = e t 1 x 3 (t) = e t It turns out that x = P(t)x always has a fundamental set of solutions. Theorem Let e 1 = (1, 0,..., 0) T, e 2 = (0, 1, 0,..., 0) T,..., e n = (0,..., 0, 1) T. Let x 1,..., x n be solutions of x = P(t)x that satisfy the initial conditions x 1 (t 0 ) = e 1,..., x n (t 0 ) = e n, t 0 I Then x 1,..., x n form a fundamental set of solutions. Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

9 Linear n th order ODEs The IVP for the linear n th order ODE is given by { y (n) + p 1 (t)y (n 1) p n 1 (t)y + p n (t)y = g(t), y(t 0 ) = y 0, y (t 0 ) = y 1,..., y (n 1) (t 0 ) = y n 1. (10) Corollary If the functions p 1 (t),..., p n (t), and g(t) are continuous on the open interval I = (α, β), then there exists exactly one solution y = z(t) of the initial value problem (10). This solution exists throughout the interval I. Corollary Let y 1,..., y n be solutions of y (n) + p 1 (t)y (n 1) p n 1 (t)y + p n (t)y = 0 (11) on I in which p 1 (t),..., p n (t) are continuous. If for some point t 0 I, the Wronskian W [y 1,..., y n ] 0, then any solution y = z(t) of (11) can be written as a linear combination of y 1,..., y n, z(t) = ĉ 1 y 1 (t) ĉ n y n (t), where the constants ĉ 1,..., ĉ n are uniquely determined. Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10

10 Homework Homework: Section 6.2 5, 9, 11 Konstantin Zuev (USC) Math 245, Lecture 31 April 4, / 10