Section4_4Solutions

Section4_4Solutions - Mathematics 421 Solutions 4.4 Spring...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ² .e t sin t/ or e t ² .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 ² .s 2 ± 2s C 2/ ± 1 ³ D ± . ± 1/.s 2 ± 2s C 2/ ± 2 .2s ± 2/ D 2.s ± 1/ .s 2 ± 2s C 2/ 2 : Using the second expression, the first step produces ˇ f t sin t g D ± d ds ² .s 2 C 1/ ± 1 ³ D ± .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/29/2009 for the course 650 421 taught by Professor Bumby during the Spring '08 term at Rutgers.

Page1 / 4

Section4_4Solutions - Mathematics 421 Solutions 4.4 Spring...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online