This assignment focuses on Laplace transforms, their basic properties, including the application to discontinuous functions, and their use for solving differential equations. 1. Basic properties of Laplace transforms: Show all of your steps. You can use tables of Laplace transforms to assist with the calculations. (a) Using a table of Laplace transforms, evaluate L {2t3 − 3e−2t }. (b) Evaluate the Laplace transform of t2 sin(bt). ∫∞ Hint: First verify the identity 0 e−st t2 f (t)dt = d2 ds2 ∫∞ 0 e−st f (t)dt. 2. Inverse Laplace transforms: Show all of your steps. You can use tables of Laplace transforms to assist with the calculations. { } (a) Evaluate the inverse Laplace transform, L−1 3s−3 + ss+2 2 +9 . Clearly, show your steps. { 1 } (b) Evaluate the inverse Laplace transform, L−1 s3 +5s . Hint: You will need a partial fraction expansion. { } 1 (c) Evaluate the inverse Laplace transform, L−1 (s+3) . 4 3. Some properties of Laplace transforms: { } . Show that it equals −F (s) − s dF . (a) Take the Laplace transform L t df dt ds (b) Evaluate the Laplace transform of the function f (t/c) where c is a constant, i.e., determine L {f (t/c)}. Begin with the integral definition of the Laplace transform. 4. Heaviside function, { H(t) = 0 if t < 0 1 if t ≥ 0 (1) (a) Find the Laplace transform of f (t) = H(t) − H(t − 3). (b) Using the integral definition, evaluate the Laplace transform of tH(2 − t) where H(·) is the Heaviside step function. 1 5. Consider a second-order linear ODE that includes a damping term: d2 y dy dy + 4 + 3y = H(t − 3) with y(0) = 0, (0) = 1. 2 dt dt dt (2) Here we lead you through the various steps in the solution via Laplace transforms. (a) Take the Laplace transform and show that Y (s) is given by Y (s) = 1 e−3s + . (s + 1)(s + 3) s(s + 1)(s + 3) (3) Which term represents the homogeneous solution to the ODE? Which term represents the particular solution? (b) Verify the partial fraction expansion 1 A B C = + + s(s + 1)(s + 3) s (s + 1) (s + 3) where you should find A, B and C. (c) Using the result from (b), take L−1 { 1 s(s+1)(s+3) (4) } . (d) Now evaluate y(t) by using the properties of the Heaviside function and taking the inverse Laplace transform of equation (3). Recall from lecture that Hc (t)f (t−c) = H(t − c)f (t − c) = L−1 {e−sc F (s)}. Hint: In the absence of forcing the solution would be the homogeneous solution. 6. Laplace transforms and differential equations: Use the method of Laplace transforms to solve the following ODEs, i.e. first take the Laplace transform to convert the ODE to an algebraic equation, then invert the expression using tables of Laplace transforms and so determine the unknown function. (a) d2 y + y = sin t + δ(t − π); dt2 (b) d2 y dy + y = e−t + 3δ(t − 3); + 2 dt2 dt y(0) = 0, y ′ (0) = 0. y(0) = 0, y ′ (0) = 3. 2 f (t) 1 tn (n = integer) ekt cos ωt sin ωt cosh ωt sinh ωt f (ct) ekt f (t) tn f (t) df dt d2 f dt2 F (s) 1 s n! n+1 s 1 s−k s 2 s + ω2 ω 2 s + ω2 s 2 s − ω2 ω 2 s − ω2 1 (s) F c c F (s − k) (−1)n (s > 0) (s > 0) (s > k) (s > 0) (s > 0) (s > |ω|) (s > |ω|) (c > 0) dn F (s) dsn sF (s) − f (0) s2 F (s) − sf (0) − H(t − a)f (t − a) e−as s −as e F (s) δ(t − a) e−as H(t − a) restrictions on s df (0) dt (s > 0) (a > 0) Table 1: able of common Laplace transforms. Reference: An excellent table of Laplace transforms is given in Chapter 29 of the Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, Dover Publications, 1965. 3
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