de4 - Dierential Equations Test 4 Solutions Answer FOUR...

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Differential Equations Test 4 Solutions Answer FOUR questions. Show all working clearly. 1. Define the Laplace transform Y = L ( y ) of y . From the definition, calculate that if y ( t ) = te at then Y ( s ) = 1 / ( s - a ) 2 . 2. (i) Show that if y ( t ) has Laplace transform Y ( s ) then y ( t ) has Laplace transform sY ( s ) - y (0). (ii) Assuming that L (cos αt ) = s/ ( s 2 + α 2 ) (or otherwise) calculate L (sin αt ). 3. By Laplace transform methods, solve the initial value problem y + 2 y + y = 2 cos t ; y (0) = 0 , y (0) = 0 . 4. By Laplace transform methods, solve the initial value problem y + 4 y = 16 te - 2 t ; y (0) = 1 , y (0) = 0 . 5. Find a solution to the differential equation ty + y = 1 using Laplace transform methods. Notice that the differential equation does not have constant coefficients; recall that multiplication by - t corresponds to differentiation by s . Solutions : 1. By definition, Y ( s ) = 0 e - st y ( t )d t. When y ( t ) = te at it is plain that Y ( a ) is undefined, while if s = a then (integrating by parts) Y ( s ) = 0 te ( a - s ) t d t = te ( a - s ) t / ( a - s ) 0 - 0 { e ( a - s ) t / ( a - s ) } d t so that Y ( s ) = te ( a - s ) t / ( a - s ) - e ( a - s ) t / ( a - s ) 2 0 ;
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