Unformatted text preview: Question 1. A function f deﬁned on R is periodic with period T > 0 if f ( t + T ) = f ( t ) for all t in R . Sketch the graph of the sawtooth function g ( t ) = t , 0 ≤ t < 1, g ( t +1) = g ( t ), over the interval [0 , 4]. Question 2. Find the Laplace transform of a periodic function f ( t ) with period T . Hint: L{ f ( t ) } = lim A →∞ Z A est f ( t ) dt = Z T est f ( t ) dt + lim A →∞ Z A T est f ( t ) dt, and then make the change of variable t = ξ + T on the last integral. Question 3. Find the Laplace transform of the sawtooth function g in Question #1. Question 4. Find the inverse Laplace transform of F ( s ) = 2 (1eπs )( s 2 + 1) . Hint: could you use the geometric series somehow? Question 5. Use the Laplace transform to solve the initial value problem u 00 + 3 u + 2 u = g ( t ) , u (0) = 0 , u (0) = 0 , where g is the sawtooth function from Question #1....
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 Fall '08
 DALLON
 Math, Differential Equations, Equations, Laplace

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