hw10 - MA341 Homework 10 Solution 7.5 9 z 00 5 z 6 z = 21 e...

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Unformatted text preview: MA341 Homework 10 Solution 7.5 9. z 00 + 5 z- 6 z = 21 e t- 1 ; z (1) =- 1, z (1) = 9 (1) To use the method of Laplace transforms, we first move the initial conditions to t = 0. This can be done by setting y ( t ) = z ( t + 1). Then, y ( t ) = z ( t + 1) , y 00 ( t ) = z 00 ( t + 1) . Replacing t by t + 1 in the differential equation in (1), we have: z 00 ( t + 1) + 5 z ( t + 1)- 6 z ( t + 1) = 21 e ( t +1)- 1 . (2) Substituting y ( t ) = z ( t + 1) in (2), we transform the initial value problem (1) into y 00 ( t ) + 5 y ( t )- 6 y ( t ) = 21 e t ; y (0) =- 1 , y (0) = 9 . (3) Next, we apply the Laplace transform to the differential equation in (3): L{ y 00 + 5 y- 6 y } = L{ 21 e t } . Using the linearity of L and applying the transform to the exponential function, we write: L{ y 00 } + 5 L{ y } - 6 L{ y } = 21 s- 1 . (4) Now let Y ( s ) := L{ y } ( s ). Applying the Laplace transform to the first and second derivatives: y and y 00 , and taking into account the initial conditions in (3), we find: L{ y } ( s ) = sY ( s )- y (0) = sY ( s ) + 1 , L{ y 00 } ( s ) = s 2 Y ( s )- sy (0)- y (0) = s 2 Y ( s ) + s- 9 . Then, substituting these expressions into (4) and solving for Y ( s ), we get: [ s 2 Y ( s ) + s- 9] + 5[ sY ( s ) + 1]- 6 Y ( s ) = 21 s- 1 ( s 2 + 5 s- 6) Y ( s ) =- s 2 + 5 s + 17 s- 1 Y ( s ) =- s 2 + 5 s + 17 ( s- 1)( s 2 + 5 s- 6) Y ( s ) =- s 2 + 5 s + 17 ( s- 1) 2 ( s + 6) . The expansion of Y ( s ) into partial fractions has the form:- s 2 + 5 s + 17 ( s- 1) 2 ( s + 6) = A s- 1 + B ( s- 1) 2 + C s + 6 . (5) Solving for the numerators, we eventually obtain: A = 0, B = 3, and C =- 1. Substitution of these values into (5) yields: Y ( s ) = 3 ( s- 1) 2- 1 s + 6 . Finally, using the tables of the Laplace transform we obtain: y ( t ) = 3 te t- e- 6 t . (6) Since z ( t + 1) = y ( t ), then z ( t ) = y ( t- 1). Hence, replacing t by t- 1 in (6) we find the solution of the IVP (1): z ( t ) = 3( t- 1) e t- 1...
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This homework help was uploaded on 04/16/2008 for the course MA 341 taught by Professor Schecter during the Spring '08 term at N.C. State.

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hw10 - MA341 Homework 10 Solution 7.5 9 z 00 5 z 6 z = 21 e...

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