Laplace Transform and Sin T Cos
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,
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,