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Material Type: Exam; Class: Differential Equations; Subject: Mathematics; University: Colorado School of Mines; Term: Spring 2008;
Typology: Exams
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MATH 225 - June 12, 2008 NAME: Exam IV - 50 minutes - 50 Points
In order to receive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all reasoning and work is provided. When applicable, please enclose your final answers in boxes.
6
f
t
Determine an expression for f (t) using step-functions.
(a) f (t) = te−^2 t^ + 2t^2 + 4
(b) f (t) = u 4 (t)e^3 t
(c) f (t) =^12 t sin(2t)
(a) F (s) =
(s − 1)(s + 1)^2
(b) F (s) = 3 s^ −^8 s^2 − 4 s + 13
(c) F (s) =^5 s^ −^2 s^2 + 4
Function f (t) Laplace transform F (s) Function f (t) Laplace transform F (s)
f ′(t) sF (s) − f (0) eat^
s − a
f ′′(t) s^2 F (s) − sf (0) − f ′(0) cos kt
s s^2 + k^2
∫ (^) t
0
f (τ ) dτ F^ (s) s
sin kt k s^2 + k^2
eatf (t) F (s − a) tnf (t) (−1)n^ d
nF dsn
uc(t) f (t − c) e−csF (s) tneat^ n! (s − a)n+
∫ (^) t
0
f (τ )g(t − τ ) dτ F (s)G(s) eat^ cos kt
s − a (s − a)^2 + k^2
s
eat^ sin kt k (s − a)^2 + k^2
t 1 s^2
uc(t) e
−cs s
tn^ n! sn+^
δ(t − t 0 ) e−st^0