lecture22 - Differential Equations & Linear...

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Unformatted text preview: Differential Equations & Linear Algebra Lecture 22 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Fundamental matrices Contents Fundamental matrices Nonhomogeneous linear systems Method of diagonalization Method of undetermined coefficients Method of variation of parameters Fundamental matrices The structure of the solutions of systems of linear differential equations can be further illuminated by introducing the idea of a fundamental matrix. Suppose that x (1) ( t ) , · · · , x ( n ) ( t ) form a fundamental set of solutions for the system x = P ( t ) x (1) on some interval α < t < β . Then the matrix Ψ ( t ) = x (1) 1 ( t ) · · · x ( n ) 1 ( t ) . . . . . . x (1) n ( t ) · · · x ( n ) n ( t ) (2) is said to be a fundamental matrix for system (17). Example Note that a fundamental matrix is nonsingular since its columns are linearly independent vectors. Example Find a fundamental matrix for the system x = 1 1 4 1 x . Solution In the last lecture, we have found that x (1) ( t ) = 1 2 e 3 t , x (2) ( t ) = 1- 2 e- t are linearly independent solutions. Thus a fundamental matrix for the system is Ψ ( t ) = e 3 t e- t 2 e 3 t- 2 e- t . Solution of initial value problems The solution of an initial value problem can be written very compactly in terms of a fundamental matrix. The general solution of system (17) is x = c 1 x (1) ( t ) + · · · + c n x ( n ) ( t ) (3) or, in terms of Ψ ( t ) , x = Ψ ( t ) c , (4) where c is a constant vector with arbitrary components c 1 , · · · , c n . When prescribed an initial condition x ( t ) = x (5) to the system (17), the vector c in (20) must be chosen to satisfy Solution of initial value problems Ψ ( t ) c = x . (6) Therefore, since Ψ ( t ) is nonsingular, c = Ψ- 1 ( t ) x (7) and x = Ψ ( t ) Ψ- 1 ( t ) x (8) is the solution of the initial value problem (17),(21). It should be emphasized, however, that to solve an initial value problem, one would ordinarily solve equation (22) by row reduction and then substitute c into (20), rather than compute Ψ- 1 ( t ) and use equation (24). Example Example For the system x = 1 1 4 1 x , find the fundamental matrix Φ such that Φ (0) = I . Solution The columns of Φ are solutions that satisfy the initial conditions x (1) (0) = 1 , x (2) (0) = 1 . Since the general solution is x = c 1 1 2 e 3 t + c 2 1- 2 e- t , we can obtain x (1) by choosing c 1 , c 2 satisfying c 1 1 2 + c 2 1- 2 = 1 . Example Solving the equation, we have c 1 = c 2 = 1 2 , thus x (1) = 1 2 1 2 e 3 t + 1 2 1- 2 e- t . Similarly we obtain x (2) = 1 4 1 2 e 3 t- 1 4 1- 2 e- t . Therefore, Φ ( t ) = 1 2 e 3 t + 1 2 e- t 1 4 e 3 t- 1 4 e- t e 3 t- e- t 1 2 e 3 t + 1 2 e- t is the fundamental matrix satisfying Φ (0) = I . Coupled and uncoupled systems The basic reason why a system of linear (algebraic or differential) equations presents some difficulty is that the equations are usually coupled . In other words, some or all of the equations involve more...
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lecture22 - Differential Equations & Linear...

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