This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Name:
Section/Time of lecture:
Professor/GSI: MATH 216 WINTER 2009
FINAL EXAM To get full score you need to carefully explain what you did. No calculators allowed.
LAPLACE TRANSFORM TABLE IS ON THE SECOND PAGE 1 Problem Points Score
1
30
2
20
3
26
4
30
+
2
2
TOTAL
108
TABLE OF LAPLACE TRANSFORMS
L( f (t)) = F (s)
n!
L ( t n ) = n +1
s
a
L(sin at) = 2
s + a2
s
L(cos at) = 2
s + a2
1
L(e at ) =
s−a
at
L(e f (t)) = F (s − a) L(u(t − a) f (t − a)) = e−as F (s)
L(δa (t)) = e−as
L(−t f (t)) = F (s)
t
L
f (τ )dτ
= F (s)/s
0 L ( f ( n ) ) = s n F ( s ) − s n −1 f (0 ) − s n −2 f (0 ) − · · · − s f n −2 (0 ) − f n −1 (0 ) 2k 3
( s2 + k 2 )2
2ks
L(t sin kt) =
2 + k 2 )2
(s L(sin kt − kt cos kt) = 2 Problem 1.
a10pt Solve t dy
sin t
+ 2y =
, y(π ) = 1.
dt
t 3 b10pt Find a (particular) solution for y − (5/t)y + (5/t2 )y = 8t3 , given that the functions y1 (t) = t and y2 (t) = t5 solve the corresponding homogeneous differential equation. 4 c10pt Find the general solution for y(4) − 3y − 4y = 0 5 Problem 2.
a10pt Solve y − 4y + 3y = 2et , y(0) = 3, y (0) = 6. 6 b10pt Solve the following system of ordinary differential equations
x = 5 x + 4y
y = −4 x + 5y 7 Problem 3.
a6pt Consider the following system
dx /dt = x2 − y + 1 dy/dt = xy − y2 + 2. Given that (1, 2) is a critical point of the system, discuss its stability. 8 b6pt Let mx = −kx + β x3 be the equation for a spring, m = k = β = 1. Write this as a ﬁrst
order system and ﬁnd the critical points ( x, y). x = y
y = · · · 9 c10pt Calculate the Laplace transform F (s) of the function f (t) when 0 if t < 2, f (t) = 4 if 2 ≤ t < 4 t if t > 4. 10 Problem 4.
a10pt Find the inverse Laplace transform of F (s) = 11 11
.
s2 s +1 b10pt Use Laplace transform to solve y − 2y + 2y = 0, y(0) = 0, y (0) = 1. 12 c10pt Solve the initial value problem y + y = f (t), y(0) = y (0) = 0 where f (t) = 13 0,
1, t<2
.
t≥2 ...
View
Full
Document
This note was uploaded on 01/30/2012 for the course MATH 216 taught by Professor Stenstones? during the Fall '07 term at University of Michigan.
 Fall '07
 Stenstones?
 Math

Click to edit the document details