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Unformatted text preview: 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 00 + 2 y + y = 2 cos t ; y (0) = 0 , y (0) = 0 . 4. By Laplace transform methods, solve the initial value problem y 00 + 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 ) = Z ∞ e- st y ( t )d t....
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This note was uploaded on 11/12/2011 for the course MAP 2302 taught by Professor Tuncer during the Fall '08 term at University of Florida.
- Fall '08