{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

SampleTest1v2 - L f = 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 2 2 s s...

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

View Full Document Right Arrow Icon
MATH3705 Solution to Test 1 Summer 2006 1 SOLUTION TO TEST 1 MATH3705, Summer 2006 1. (2 marks) Express the following rational function as a sum of partial fractions: F ( s ) = ) 1 )( 1 ( 2 + - s s s . Solution . F ( s ) = [1 / ( s - 1) - s / ( s 2 + 1) + 1 / ( s 2 + 1)] / 2. 2. (2 marks) Find the inverse Laplace transformation of F ( s ) = 1 / ( s - 1) - s / ( s 2 + 1) + 1 / ( s 2 + 1). Solution . L [ F ( s )] = e t - cos t + sin t . 3. (5 marks) Use the definition of Laplace transformation to find the Laplace transformation of the function f ( t ) = t e t . Solution . L [ f ] = ( 1) ( 1) ( 1) 0 0 0 0 1 1 1 1 t st s t s t s t te e dt t de e dt te s s - - - - - - - = - = - - - ( 1) ( 1) 2 2 0 0 1 1 1 1 ( 1) ( 1) s t s t e dt e s s s - - - - = = - = - - - . 4. (5 marks) Find the Laplace transformation of the function = + < + = + < = ... , 2 , 1 , 0 , 2 2 1 2 0 ... , 2 , 1 , 0 , 1 2 2 1 ) ( k k t k k k t k t f The graph of this function is as follows: Solution . This function has a fundamental period 2.
Image of page 1

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

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

Unformatted text preview: L [ f ] = [ ] ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 1 1 ) ( 1 1 2 1 2 1 2 2 2 s s s st s st s st s e s e s e e e s dt e e dt e t f e---------+ =--=--=-=-∫ ∫ . 5. (6 marks) Use Laplace transformation to solve the differential equation t f ( t ) MATH3705 Solution to Test 1 Summer 2006 2 y" + y = e t , y (0) = 0, y' (0) = 1. Solution . Let Y ( s ) = L [ y ]. Take the Laplace transformation on both sides: s 2 Y-1 + Y = 1 / ( s-1). ( s 2 + 1) Y = s / ( s-1). Y = s / [( s-1)( s 2 + 1)] = (1 / 2)(1 / ( s-1) -s / ( s 2 + 1) + 1 / ( s 2 + 1)). Take the inverse Laplace transformation, using the result in Question 1 and 2, y = (1 / 2)( e t-cos t + sin t )....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern