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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.
 Spring '08
 Schecter
 Differential Equations, Equations

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