Chapter 5
Exact
First-Order Differential
Equations
Recall
Implicit Differentiation
7 xy 4 y 2 xe 13
2
3y
dy
dy
7 y 14 xy 4
dx
dx
2
dy
2e 6 xe
0
dx
3y
3y
Recall
Partial Derivatives
f x, y 7 xy 4 y 2 xe
2
3y
f
2
3y
7 y 2e
x
f
3y
14 xy 4 6 xe
y
Whats the c
Math 271
More Applications
of
First-Order, Diff. Eq.
Name that Technique
Which technique for the following diff. eq.?
2 xy 6 x 2 dx x 2 8 y dy 0
2 xy 6 x
y 2
x 8y
tx
2
8 ty t tx 8 y
2
2
Exact
2x 2x
Zombies
Suppose a student returns to college
with the
Section 6.3
RLC Ciruit
Kirchhoffs 2nd Law
The impressed voltage,
E(t), on a closed loop
must equal the sum of
the voltage drops in the
loop.
Circuit
i is current, q is charge
Resistor
iR
Ohms ()
Capacitor
i
q
C
E (t )
Volts (v)
di
Inductor L
dt
henries (h
Math 271
Power Series Solutions of
Linear Differential
Equations with Variable
Coefficients
Power
Man
Power Series Warm-Up 1
1
n
w
1 w n 0
We have
Whats the power series for
x
1 x4
n
x
x 2 x 4 x 4n2
1 x4
n 0
n 0
2
2
Power Series Warm-Up 2
We have
1
1 x
Solve y xy xy 2 .
w ke
This is a Bernoulli equation due to the y 2
multiplier on the right side of the equation.
We have n = 2, so we use the substitution:
1 2
x
2
1 where k ec .
Now that we have a solution for w, we need
to substitute back to get a solu
Modeling
Pop Quiz
What is
wrong
with this
picture?
Pop Quiz 2
Two students
start a rumor
about Math 271.
Which of the six
models
represents how
the rumor is
spread (the
number of
students who
know the rumor)?
M = total population of the college
Pop Quiz 2
Separable Equations
Separable Equations
Various Forms
A x dx B( y)dy 0
dy
g x p( y )
dx
h( y)dy g x dx
Separable Equations
Another common form:
dy
g x p( y )
dx
dy
g x dx
p( y )
Steps for Separable Diff. Eq.
1. Identify
2. Separate (if needed)
3. Inte
Section
5.4
The method of
undetermined coefficients
y p Ax
Example 0
Find the general solution to
y y 5
y c1e
1 x
c2e c1e
y c1e
0x
1 x
1 x
c2
c2 5 x
y p Ae
Example 1 (5.4.1)
Find the general solution to
y 7 y 12 y 4e
2x
4 Ae 14 Ae 12 Ae 4e
2x
2x
2x
y c
Section 6.2
Applications of 2nd order
linear differential equations
Hookes Law
General Formula for Spring
my cy ky F
External
Force
mass
Damping
Force
Spring
Constant
Changed
orientation
Example 6 Damping Motion
A mass weighing 16 pounds is
attached to a
Math 271 Section 9.3
Undetermined Coefficients for Higher Order Equations when F t et G t .
Steps:
1. Find the homogeneous solution by solving the characteristic equation.
2. Use y p uet for the particular solution and find the derivatives.
3.
4.
5.
6.
7.
Section 6.1
Applications of 2nd order
linear differential equations
Hookes Law
Authors Perspective
Damping
Force
F cy
Gravity
mg
+
direction
L
L
Spring
Force
FS k L
At any
time
-direction
Example 1: Free, Undamped
A mass weighing 2
pounds stretches a
spri
Another
Substitution
A different kind of
Homogeneous Nonlinear Equation
What is this?
y xy
y
2
x
2
Label:
First Order, nonlinear,
homogeneous
the (y/x) version
What is this?
y xy y
xy
y
2 2
2
x
x
x
2
First order,
Nonlinear
Homogeneous
because in
form o
My brain is
exploding!
Section
5.5
More methods of
undetermined coefficients
General Rules
1. If g(t) has an exponential, list it
automatically. Take a guess on the other
stuff.
2. Products of poly and trig: write down a
guess for the poly and write it wi
Math 271
Pierre Laplace
Laplace Transforms
Functions and Transforms
Whats a function?
Whats a mathematical transformation?
Starbucks Investment
Whats the growth rate from 7/1993 to
7/2006?
Log Transform
m = 0.24162
= 27%
July 1993 $1.48
July 1993
0.39204
Play it again Sam
Section 9.2
Higher order Linear
Homogeneous Differential
Equations
with
constant coefficients
Section 9.1 Key Points
Key Points
Solutions to differential equation
need to be linearly independent
Solutions need to satisfy the
Wronskian
Basic Concepts
Grouping Activity
dy
y2 x 2x
dx
y x 2
d2y
dy
x y0
2
dx
dx
d3y
dy
x 1 x y sin y
dx3
dx
dy
x2
dx
dy
2 xy x 2 x
dx
10 y y e x
sin x y 2
y
x
d2y
2 xy
2
dx
Order
Background:
Whats the degree of a function?
y 3x 21x 18
6
3
Highest power in
Math 271
Applications
of
First-Order, Linear Diff. Eq.
Name that Technique
Which technique for the following diff. eq.?
4xyy y x
2
2
Bernoullis
2
y2
4 xyy
x
4 xy
4 xy 4 xy
y
x
y
4x 4 y
1
x 1
y
y y
4x
4
Bernoulli Family
Population Growth
On a trip to Ma
Math 271
Identifying First Order Differential Equations
Spring 2014
Classify each differential equation as separable, exact, linear in terms of y, linear in terms of x, homogeneous,
or Bernoulli. Some equations may be more than one kind.
M x, y dx N x, y
Bernoulli
Equations
First Order Linear Differential
Equations with a Twist
The Twist
Start with
y p x y q( x)
What happens when there is a
y-term on right side?
y p x y q ( x) y
n
Some power
Not 1, not 0
Example
xy y xy
3
Example
First adjustment:
get y a
Fields
Starting with graphs, getting slopes
Which is the
correct graph of
y t dt
2
A B CDE
Solutions to the Diff. Eq.
Slope field for
y t
2
Slope field for
y y t
Slope field for
y y 1 y
Math as Art
More Art
http:/calvinshobbies.blogspot.com/2011/01/slope
Play it again Sam
Section 9.2
Higher order Linear
Homogeneous Differential
Equations
with
constant coefficients
Practice 1
Solve the following IVP:
3
y
5 y 22 y 56 y 0
y 0 1, y 0 2, y 0 4
To find the k-values, solve
3
2
k 3k 4 0
k 1 k 2
2
0
Practice 1
Math 271
Solutions of Linear
Systems by Laplace
Transforms
Good old days
Example 1
Solve
Laplace
w 2 z 0
w y z 2 x
w 2 y z 0
w 0 0, w 0 0
z 0 1, z 0 0, y 0 0
Same Process for a System
Step 1: Transform differential equation
Step 2: Solve for Y, U, V, Z,
S
Section 8.3
Laplace Transform for
2nd order differential
equations
Shifts happen
L e
4t
sin 6t e e
st 4t
0
e
0
s 4 t
sin 6t dt
sin 6t dt
Shift left of 4
What about?
L e
4 t
sin 6t
L e
4 t
6
s 4
cos 6t
Multiplying by
e
kt
2
36
s4
s 4
2
is like a sh
First Order
Linear ODE
Variation of Parameters
BARNHARDT (slowly, thoughtfully):
Yes - that will reproduce the firstorder terms. But what about the effect
of the other terms?
KLAATU: Almost negligible. With
variation of parameters, this is the
answer.
Lab
Section 9.4
Variation of Parameters
Of Higher Order
Differential Equations
Example 1
Solve
the setup
y 5 y 25 y 125 y 1000
Given
yh c1e c2 cos 5t c3 sin 5t
5t
The variation
y p u1e u2 cos 5t u3 sin 5t
5t
Example 1 making the system
For three components of