MATH251-Worksheet 4

MATH251-Worksheet 4 - MATH 251 Work sheet / Things to know...

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© 2011 Zachary S Tseng 1 MATH 251 Work sheet / Things to know Chapter 7 1. System of first order linear differential equations Standard form: What makes it homogeneous? We will be working with systems whose equations have constant coefficients only. Know the matrix-vector shorthand form of a system of linear equations. What does each part mean? x ′ = Ax + g Every n -th order linear equation can be converted (not uniquely) to a system of n first order linear equations. Know how this can be done. Ex. 7.1.1 Convert each linear equation into a system of first order equations. (a) y″ - 4 y′ + 6 y = 0 (b) y″′ + 5 y″ + 7 y′ - 9 y = 5 t 3 cos 2 t
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© 2011 Zachary S Tseng 2 Before proceed any further, be sure to familiarize yourself with basic matrix notations and operations, as well as knowing how to find eigenvalues and eigenvectors of a given 2-by-2 matrix. 2. The characteristic equation (again) Given a 2-by-2 coefficient matrix A , how to find its characteristic equation ? As a determinant: or, a simple formula: Any root, r , of the characteristic equation, is called an eigenvalue of the matrix A . It has the property that the vector-valued function x = k e rt , where k is a corresponding eigenvector of r , always satisfies the homogeneous system of first order linear equations x ′ = Ax . Note that for each eigenvalue we just need one eigenvector. We can choose whichever eigenvector that is the most convenient. Ex. 7.2.1 Find the eigenvalues and corresponding eigenvectors of the matrix: - 1 2 3 4
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© 2011 Zachary S Tseng 3 Ex. 7.2.2 Rewrite the linear system below in matrix-vector notation. What is its coefficient matrix A ? Find the eigenvalues and corresponding eigenvectors of A . x 1 ′ = 2 x 1 + 5 x 2 x 2 ′ = 3 x 1 + 4 x 2 Ex. 7.2.3 Rewrite the linear system below in matrix-vector notation. What is its coefficient matrix A ? Find the eigenvalues and corresponding eigenvectors of A . x 1 ′ = 5 x 1 + 4 x 2 x 2 ′ = ±2 x 1 + 1 x 2 [ In this example the eigenvalues and eigenvectors are complex numbers / vectors. See section 4 below for the case where the coefficient matrix has complex eigenvalues and eigenvectors. ]
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© 2011 Zachary S Tseng 4 3. The general solution of a system of 2 first order linear equations Structure of a general solution: For a homogeneous linear system of 2 equations: x = C 1 x 1 + C 2 x 2 The fundamental solutions x 1 and x 2 are determined by the eigenvalues and eigenvectors of the coefficient matrix A of the system. (If the system should be nonhomogeneous, then the expression above forms the complementary part of the general solution. You would then need to find a nonhomogeneous solution to complete the general solution.) Similar to what we have experienced while solving second order linear equations, the general solution takes on different forms depending on the nature of the roots of the characteristic equation (that is, the eigenvalues of the system’s coefficient matrix). General solution:
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MATH251-Worksheet 4 - MATH 251 Work sheet / Things to know...

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