MIT18_03S10_ls6

MIT18_03S10_ls6 - LS.6 Solution Matrices In the literature,...

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Unformatted text preview: LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions and results for a 2 2 system, but they generalize immediately to n n systems. 1. Fundamental matrices. We return to the system (1) x = A ( t ) x , with the general solution (2) x = c 1 x 1 ( t ) + c 2 x 2 ( t ) , where x 1 and x 2 are two independent solutions to (1), and c 1 and c 2 are arbitrary constants. We form the matrix whose columns are the solutions x 1 and x 2 : x 1 x 2 (3) X ( t ) = ( x 1 x 2 ) = . y 1 y 2 Since the solutions are linearly independent, we called them in LS.5 a fundamental set of solutions, and therefore we call the matrix in (3) a fundamental matrix for the system (1). Writing the general solution using X(t) . As a first application of X ( t ), we can use it to write the general solution (2) eciently. For according to (2), it is x = c 1 x 1 + c 2 x 2 = x 1 x 2 c 1 , y 1 y 2 y 1 y 2 c 2 which becomes using the fundamental matrix c 1 (4) x = X ( t ) c where c = , ( general solution to (1) ) . c 2 Note that the vector c must be written on the right, even though the c s are usually written on the left when they are the coecients of the solutions x i . Solving the IVP using X(t) . We can now write down the solution to the IVP (5) x = A ( t ) x , x ( t ) = x . Starting from the general solution (4), we have to choose the c so that the initial condition in (6) is satisfied. Substituting t into (5) gives us the matrix equation for c : X ( t ) c = x . Since the determinant | X ( t ) | is the value at t of the Wronskian of x 1 amd x 2 , it is non-zero since the two solutions are linearly independent (Theorem 5.2C). Therefore the inverse matrix exists (by LS.1), and the matrix equation above can be solved for c : c = X ( t ) 1 x ; using the above value of c in (4), the solution to the IVP (1) can now be written (6) x = X ( t ) X ( t ) 1 x . 25 26 18.03 NOTES: LS. LINEAR SYSTEMS Note that when the solution is written in this form, its obvious that x ( t ) = x , i.e., that the initial condition in (5) is satisfied....
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MIT18_03S10_ls6 - LS.6 Solution Matrices In the literature,...

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