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Homework Set 5 f(t) F(S) Section 4.1: Apply the definition to directly find the Laplace transforms of the given functions. (s > 0) 1 (s > 0) S- 1. Kt) = 12 2. f = 23t+1 Use transforms from the Table (op right) to find the Laplace transforms of the given functions. t" ( n20) (s > 0) r(a + 1) 1a (a > -1) (s > 0) 5+1 3. f(t) = VE +8t 4. f(t) = sin(2tcos(2t) Use the...
Applied Mathematics Laplace Transforms 1. Consider a smooth function f(t) defined on 0 t<o, with Laplace transform F(s) (a) Prove the First Shift Theorem, which states that Lfeatf(t)) = F(s-a), where a is a constant. Use the First Shift Theorem to find the inverse trans- form of s2 -6s 12 6 marks (b) Prove the Second Shift Theorem, which states that L{f(t-a)H(t-a))-e-as F(s), where H is the Heaviside step function and a is a positive constant. Use the First and...
Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your answer as a function of t.) ℒ−1 1/(s^2 + s − 56) Some Inverse Transforms (a) 1 = L-1 (b) " = L-1 1 n = 1, 2, 3, ... (c) eat = L-1 L-1 (d) sin kt = L-1 k 92 + k? (e) cos kt = L- 52 + k ****] ) S (f) sinh kt = ! k 92 – k (g)...
need help all those questions. 10. Solve the following systems of linear differential equations: 11. Determine the Laplace transform of each of the following functions: (a) fe)-2t+1, 0StcI , 21 (b) f(t) te (c) f(t) = cos t cos 2t (Hint: Examine cos(a ± b).) Determine the inverse Laplace transform of each function: 12. (a) F(s) = 52 +9 is Demin 13. Determine L{kt cos kt + sin kt). 0, t< a 14. Determine L(cos 2t)U(t-r), where U(t-a)={ 15. Use...
(1 point) Find the Inverse Laplace transform f(t) = --! {F(s)) of the function F(s) 120 120 f(t) = -1 help (formulas)
(1 point) Find the inverse Laplace transform f(t) = 2" (F(s)} of the function F(s) = 2s 8²-1 (t) = -1 ^{}--G-- help (formulas)
(15 points) Use the convolution theorem to find the inverse Laplace transform f(t) of F(s) = 32 2 $'(92 + 4) f(t) = 16sin^2(t)
Find the inverse Laplace transform, f(t) of the function F(s)+ f(t) Points possible: 1 S > 3 Preview t>0 Enter an algebraic expression [more..]
2t +1 if 0 <t< 2 Consider f(t) = { | 3t if t > 2. (a) Use the table of Laplace transforms directly to find the Laplace transform of f. (b) Express f in terms of the unit step function, then use Theorem 6.3.1 to find the Laplace transform of f.
3 (1 point) Find the inverse Laplace transform f(t) = --! {F(s)} of the function F(s) = - 25 32 +25 $2 + 16 f(t) = -1 e='{-6816+,725)} = help (formulas)