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Unformatted text preview: Solution of Linear Differential Equations Srinivas Palanki University of South Alabama Srinivas Palanki (USA) Solution of Linear Differential Equations 1 / 15 Exponential of a Matrix Exponential of a Matrix The quantity e at where a and t are scalar is an infinite series defined as: e at = 1 + at 1! + a 2 t 2 2! + a 3 t 3 3! + ... + a k t k k ! + ... (1) When we have a matrix A instead of a scalar a , we can form an infinite series similar to the above equation and use it as a definition of the exponential of a matrix . e At = I + At 1! + A 2 t 2 2! + A 3 t 3 3! + ... + A k t k k ! + ... (2) Srinivas Palanki (USA) Solution of Linear Differential Equations 2 / 15 Exponential of a Matrix Properties of e At 1 e At t =0 = I 2 d dt ( e At ) = A . e At = e At . A 3 A . Z t e Aj dj = Z t e Aj dj . A = e At I 4 e A ( t + s ) = e At e As 5 e At e At = e At e At = I Srinivas Palanki (USA) Solution of Linear Differential Equations 3 / 15 Linear Matrix Differential Equation Analytical Solution Using the properties of e At , we can find the solution of a linear matrix differential equation with constant coefficients....
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 Spring '10
 Dr.SrinivasPalanki

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