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Section4_4Solutions

Section4_4Solutions - Mathematics 421 Solutions 4.4 Spring...

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Mathematics 421 Solutions 4.4 Spring 2008 Introduction Exercises 10 and 14 involved many steps in a single problem, so there were few solutions that were completely correct. These solutions are presented as an example of how the work on such problems may be organized to improve the chance of having a correct answer at the end of your effort. Problem 10 Solve y 0 y D te t sin t y.0/ D 0 The left side Introduce Y for the transform of y to get ˇ f y g D Y ˇ ˚ y 0 D sY ˇ ˚ y 0 y D .s 1/Y The right side To find the Laplace transform of te t sin t , we may think of the expression as either t .e t sin t/ or e t .t sin t/ . These lead to two different computations that reach the same value. In both cases, we start with ˇ f sin t g D 1 s 2 C 1 : Using the first expression, the function of t is multiplied by e t , which requires that s be replaced by s 1 in the transform. Thus, ˇ ˚ e t sin t D 1 .s 1/ 2 C 1 D 1 s 2 2s C 2 : The transform of the result of multiplication by t is the negative of the derivative of the basic transform. Thus, ˇ ˚ te t sin t D d ds .s 2 2s C 2/ 1 D . 1/.s 2 2s C 2/ 2 .2s 2/ D 2.s 1/ .s 2 2s C 2/ 2 :

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Section4_4Solutions - Mathematics 421 Solutions 4.4 Spring...

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