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5 Questions of Differential Equations - Practice Exam | MATH 225, Exams of Differential Equations

Material Type: Exam; Class: Differential Equations; Subject: Mathematics; University: Colorado School of Mines; Term: Spring 2008;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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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.
1. (5 Points) Calculate the Laplace transform of f(t) = te2tusing the integral definition.
2. (5 Points) Given the following graph of f:
6
-
f
t
Determine an expression for f(t) using step-functions.
1
pf3
pf4

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Download 5 Questions of Differential Equations - Practice Exam | MATH 225 and more Exams Differential Equations in PDF only on Docsity!

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.

  1. (5 Points) Calculate the Laplace transform of f (t) = te−^2 t^ using the integral definition.
  2. (5 Points) Given the following graph of f :

6

f

t

Determine an expression for f (t) using step-functions.

  1. (10 Points) Given f (t) find F (s):

(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)

  1. (10 Points) Given F (s) find f (t):

(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