10-21-review-soln - Differential Equations Fall 2011...

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Unformatted text preview: Differential Equations Fall 2011 Friday, 10/21/11 In-class review for Section 7.6 Solutions 1 Express the piecewise defined function f ( t ) = 2 < t < 3 , 3 < t < 7 , t 5 7 < t < 9 , sin t t > 9 in a single line in terms of Heaviside functions. Solution: It is just 2- 2 u ( t- 3) + t 5 u ( t- 7)- t 5 u ( t- 9) + (sin t ) u ( t- 9) . 2 Find the Laplace transform of the solution y to the initial value prob- lem 2 y - 3 y + y = braceleftbigg t 2 < t < 5 , sin t t > 5 y (0) = 1; y (0) = 0 . Solution: To save some writing, let Y ( s ) = L { y } . For the lefthand side, we have L { y } = Y, L { y } = sY- y (0) = sY- 1 , L { y } = s ( L { y } )- y (0) = s 2 Y- s. So, we get L braceleftbig 2 y - 3 y + y bracerightbig = 2( s 2 Y- s )- 3( sY- 1)+ Y = (2 s 2- 3 s +1) Y +(- 2 s +3) . To transform the righthand side, we first need to express it in terms of Heaviside functions. We have that braceleftbigg t 2 < t < 5 , sin t t >...
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This note was uploaded on 12/10/2011 for the course MAP 2302 taught by Professor Tuncer during the Fall '08 term at University of Florida.

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10-21-review-soln - Differential Equations Fall 2011...

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