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Unformatted text preview: Solution of Linear Systems of Ordinary Differential Equations James Keesling 1 Linear Ordinary Differential Equations Consider a first-order linear system of differential equations with constant coefficients. This can be put into matrix form. dx dt = Ax (1) x (0) = C Here x ( t ) is a vector function expressed as a column vector, x : R R n and A is an n n matrix. The solution of this can be obtained by using what is called the exponential of a matrix . For the n n matrix A , define exp( t A ) = X k =0 t k k ! A k . This matrix series will converge for all values of t . For each value of t , the limit exp( tA ) is an n n matrix. Furthermore, exp(( t + s ) A ) = exp( tA ) exp( sA ) and d exp( tA ) dt = A exp( tA ) . The above features are similar to the scalar value exponential function. On the other hand, the matrix exponential function does have some surprising differences. Among these is that it may be the case that for two n n matrices A and B , it may be the case that exp( A + B ) 6 = exp( A ) exp( B ). 2 Matrix Solution of the Equation The Picard method shows that a linear system of differential equations has a unique so- lution. However, we can easily produce the solution to (1) using the matrix exponential function. The solution is obvious from the above formulas....
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- Summer '08