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Unformatted text preview: Differential Equations Assignment #5 (Brief Solutions) Laplace Transforms, and Discontinuous Forces 1. Compute the Laplace transform of each of the following functions, using the integral definition of the transform: These Laplace transforms can be checked in any readily available table of Laplace transforms. (a) f ( t ) = t F ( s ) = 1 s 2 (b) f ( t ) = sin( t ) F ( s ) = 1 1 + s 2 2. In class we discovered that we could express the Laplace transform of a derivative wholely in terms of the Laplace transform of the original function: L ( f ) = s L ( f )- f (0) Find a similar formula for the Laplace Transform of f 000 . (You may either apply integration by parts to the definition, or use the formula above recursively). Applying the above formula recursively we obtain: L ( f 000 ) = s L ( f 00 )- f 00 (0) L ( f 000 ) = s s L ( f )- f (0)- f 00 (0) L ( f 000 ) = s s [ s L ( f )- f (0)]- f (0)- f 00 (0) 1 3. Use the Laplace transform to solve each of the following non-homogeneous forcing problems: (a) y 00- y = sin( t ), y (0) = 0, y (0) = 1 Applying the Laplace transform yields the following: s 2 Y- sy (0)- y (0)- Y = 1 1 + s 2 (1- s 2 ) Y- 1 = 1 1 + s 2 Y = 1 1- s 2 + 1 (1 + s 2 )(1- s 2 ) Applying Partial fractions to this transform yields the following decomposition: Y = 1 2 1 s + 1- 1...
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