DEPARTMENT OF MATHEMATICAL SCIENCES FACULTY OF SCIENCE UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1793 DIFFERENTIAL EQUATIONS 1. TUTORIAL 3

Use the deﬁnition of Laplace transform to determine F (s) for the following functions. a. f (t) = 5e5t . c. f (t) = sinh 4t. e. f (t) = g. f (t) = t, 5, 0 4. t e , 0 < t < 2 h. f (t) = 0, 2 < t < 4 5, t > 4. f. f (t) =

sin 2t, 0 < t < π 0, t > π.

2.

Use the Laplace transform table to ﬁnd F (s) for the given function. a. f (t) = 2 sin t + 3 cos 2t. c. f (t) = 2t2 − 3t + 4. e. f (t) = e−2t sin 5t. g. f (t) = 5e2t + 7e−t . i. f (t) = t2 − t sinh t − 2e−t sin 3t. k. f (t) = te−t cos 2t. b. f (t) = e2t sinh2 t. d. f (t) = (sin t + cos t)2 . f. f (t) = sin t cos t. h. f (t) = (t − 1)2 + t sinh 2t. j. f (t) = t3 e−4t + t sin t. l. f (t) = 2t2 e−t cosh t.

3.

Sketch the graph of the given function for t ≥ 0, and ﬁnd its Laplace transform. a. f (t) = (t − 4)H(t − 4). c. f (t) = (t − 3)H(t − 1). e. f (t) = e−2t H(t − 4). b. f (t) = H(t − 2) − H(t − 3). d. f (t) = cos(t − π)H(t − π).

4.

Express the given function in terms of unit step functions, and ﬁnd its Laplace transform. 0, 0 < t < 2 t e, 0 < t < 2π t, 2 < t < 5 a. f (t) = b. f (t) = cos t, t > 2π. 2t e , t > 5. 0, 0

...7
LaplaceTransform
The Laplacetransform can be used to solve diﬀerential equations. Besides being a diﬀerent and eﬃcient alternative to variation of parameters and undetermined coeﬃcients, the Laplace method is particularly advantageous for input terms that are piecewise-deﬁned, periodic or impulsive. The direct Laplacetransform or the Laplace integral of a function f...

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...f ( t ) = L -1 {F ( s )} 1. 3. 5. 7. 9. 11. 1 t n , n = 1, 2,3,K tsin ( at ) tsin ( at ) sin ( at ) - at cos ( at ) cos ( at ) - at sin ( at ) sin ( at + b ) sinh ( at ) e at sin ( bt ) e at sinh ( bt ) t ne at , n = 1, 2,3,K uc ( t ) = u ( t - c )
Heaviside Function
F ( s ) = L {...

...Laplace Transformation Laplace transformation is a Mathematical tool which can be used to solve several problems in science and engineering. The transformed was first introduced by Pierre-Simon Laplace a French Mathematician, in the year 1790 in his work on probability theorem. Application of LaplaceTransform The Laplacetransform technique is applicable in many fields of science and technology...

...Find the Laplacetransform of the following functions 1. f (t) = t − 1, 0 < t < 3; 7, t > 3. 2. f (t) = cost − 0,
2π 3
, 0 2π . 3
2π ; 3
4, 0 < t < 1; −2, 1 < t < 3; 3. f (t) = 5, t > 3. 5. f (t) = 3t3 + e−2t + t 3 7. f (t) = cos3 2t 9. f (t)...

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