Laplace2 - The Laplace Transform We again consider the...

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The Laplace Transform We again consider the nonhomogeneous IVP ˙ y = A y + g y ( x 0 ) = y 0 . In class, following Braun pp. 368-369, we showed that the Laplace transform applied to both sides of this equation gave the new matrix equation ( sI - A ) Y ( s ) = y 0 + G ( s ) where we defined Y ( s ) = Y 1 ( s ) Y 2 ( s ) . . . Y n ( s ) = L [ y 1 ( x )] L [ y 2 ( x )] . . . L [ y n ( x )] = L y ( x ) and similarly G ( s ) = G 1 ( s ) G 2 ( s ) . . . G n ( s ) = L [ g 1 ( x )] L [ g 2 ( x )] . . . L [ g n ( x )] = L g ( x ) . Then to solve this nonhomogeneous IVP we can just (1) Apply the Laplace transform to both sides of the nonhomogeneous equation to get the new matrix equation ( sI - A ) Y ( s ) = y 0 + G ( s ) . (2) Solve for Y , the Laplace transform of the solution, by any method from linear algebra. The easiest perhaps being to multiply both sides by the inverse of ( sI - A ) to get Y ( s ) = ( sI - A ) - 1 y 0 + ( sI - A ) - 1 G ( s ) . (3) Now that the vector Y ( s ) is known, the solution y ( x ) is found by taking the inverse Laplace
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