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Unformatted text preview: Variation of Parameters We now consider the nonhomogeneous IVP ˙ y = A y + g y ( x ) = y . In class, following Braun pp. 360-361, we showed that the method of variation of parameters could be used to find the following solution to this IVP: y ( x ) = e A ( x- x ) y + Z x x e A ( x- t ) g ( t ) dt. Then to solve this nonhomogeneous IVP we can just (1) Solve the homogeneous equation to get a fundamental matrix solution X ( x ) . (2) Compute the matrix exponential by e Ax = X ( x ) X- 1 (0) . (3) Plug this, along with all of the given information, into the above formula and perform the integration component by component to find y ( x ) . I Example Solve the IVP: ˙ y = 3- 4 1- 1 y + e x e x , y (0) = 1 1 . Solution In the above notation we have A = 3- 4 1- 1 , g = e x e x ,x = 0 , y = 1 1 . We begin by finding two linearly independent solutions to the homogeneous equation ˙ y = A y so we can form a fundamental matrix solution X . We compute det( A- λI ) = (3- λ )(- 1- λ ) + 4 = λ 2-...
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- Linear Algebra, ex, Det