Math 2171 section 004
x7.2, 7.3 Laplace transforms
Professor Alan Dow
November 5, 2013
We use the letter v to denote the Laplace transform, and v1
denotes the inverse transform
the idea is the following
functions
denoted
f (t )
in the time domain
v
3
are
Math 2171 section 004
x4.2
Professor Alan Dow
October 1, 2013
Last day we checked that
if y1 and y2 are solutions to
ay HH + by H + cy = 0
then so is
y = c1 y1 + c2 y2 for any pair c1 ; c2
Also, we mentioned that ay HH + by H + cy = 0 also satises the
exi
Math 2171 section 004
3.4-3.2
Professor Alan Dow
September 22, 2013
Suggested problems: 3.4 5, 9, 12, 21;
3.2 1, 3, 5, 7, 14, 15, 19; 3.3 1, 3, 5, 7; 3.5 1,2
Example 3: a 75 kg parachutist drops from 4000m and opens
chute after 1 min. Air coecient is b1 =
Test 2 review for Oct 14 class
1. Solve the ivp: (tan(y ) 2) dx + (x sec2 (y ) + 1=y ) dy = 0 with y ( ) = 1
2. Find general solution:
dy
dx
= (2x + y 1)2
3. A 70 L tank has two input sources. The rst sends in a solution of :05 kg/L
salt-water at 2 L/min
Math2171-004 Practice for Test 1
(1) Decide if the existence and uniqueness theorem of Chapter 1 guarantees
a unique solution for each of the following ivp's. If \No" explain why,
if \Yes", determine a suitable rectangle R as per the theorem.
p
dy
(a) dx
Math 2171 section 004
Impulse and Dirac Delta
Professor Alan Dow
November 23, 2013
Rewriting a piecewise dened function f (t) in terms of Heaviside
functions u(t a)
0
t<a
f (t) a < t < b
1
e.g. f (t) =
f2 (t) b < t < c
f (t) c < t
3
is same as
f (t) = f1
Math 2171 section 004
4.4 Non-homogeneous and undetermined coes
Professor Alan Dow
October 19, 2013
review
homogeneous ay + by + cy = 0 should have two
linearly independent solutions y1 , y2 which we nd by solving the
associated auxiliary equation. The ge
Math 2171 section 004
review of 4.4-5 and cover 4.6
Professor Alan Dow
October 27, 2013
review
1. yp for y y = t cos(3t) has form
(A1 t + A0 ) cos(3t) + (B1 t + B0 ) sin(3t) because
y1 = e t , y2 = e t determine yh = c1 y1 + c2 y2
Lets nd solution with y
Math 2171 section 004
x4.4 cos and sin case and x4.5
Professor Alan Dow
October 22, 2013
A sin( t + )
a typical solution to ivp with complex roots is
ae t cos( t ) + be t sin( t )
as you know.
this can then be converted to a simpler form
e t A sin( t + )
Math 2171 section 004
test 3 review
Professor Alan Dow
November 10, 2013
Quick Summary:
Homog: ay + by + cy = 0 has solutions y1 , y2
r1 t r2 t
overdamped, discr. > 0
e , e
rt , t e rt
e
critically damped, discr = 0
t
t sin(t) underdamped, discr. < 0
e
Math 2171 section 004
exam review
Professor Alan Dow
December 1, 2013
First order: separable, linear, exact, Bernoulli, mixing
no Euler, no existence uniqueness, no advanced numerical, no
gravity
Second order: Homogeneous solution using aux. eqn., ivp sol
Math 2171 section 004
review and 4.10 Forced Mechanical
Professor Alan Dow
November 3, 2013
Summary: last day we learned about underdamped, critically
damped, overdamped versions of
my + by + ky = 0 (with all parameters 0)
The interesting thing about crit
Math2171-002 Test 3 Practice
(1) Write down a second-order constant coecient dierential equation which
has e2 and e3 as solutions. Then, for this equation, solve the ivp having
y (0) = 1 and y H (0) = 2.
t
t
(2) Solve the ivp 2y HH + 2y H + 5y = 0, y (0)