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Agenda
Introduction
Sections 1.1-1.3
Reminders
Read §1.1-1.3 & 2.1
Do problems for §1.1-1.3
O�ce hours Tues, Thurs3-4:30 pm (5852 East Hall)
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Text Book
The textbook for the class is the 11th edition of Elementary
Di↵erential Equations by Boyce, Diprima, and Meade.
This is a fast paced course and you are expected to read aheadbefore showing up to class.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Grading Policy
Your grade is determined using the following weights:
First Exam - 10%
Second Exam - 25%
Final Exam - 35%
Homework - 20%
Lab Assignments - 10%
Grades given on individual homeworks, labs, and exams willnot be “curved.” However the historical average cumulativegrade for Math 316 is about a “B,” and you should expect asimilar statistic for our class.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Exams
First Exam
Thursday, 1/23, 6-7 pm
Second Exam
Thursday, 2/27, 6-7 pm
Final Exam
Thursday, 4/26, 1:30-3:30 pm
Note that exam dates are absolutely firm. Travel plans will notbe considered as a su�cient excuse to take an exam on adi↵erent date.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Homework
Homework will be posted on Canvas after each lecture.
You should attempt the problems before the next lecture.
You should keep a notebook for writing out assignedhomework problems.
Several times throughout the term, you will be asked toturn in some problems as written homework.
Only the homework that you are asked to turn in willcount towards your grade.
To succeed in this course, you will need to understand
and be able to solve the homework problems on your own.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Labs
We will have a few classes in a computer lab.
Lab classes will be held in room 2000 of the ShapiroUndergraduate Library (unless stated otherwise).
The labs will involve the use of Mathematica for thestudy of di↵erential equations.
You will need to submit a lab write-up before the classfollowing the lab.
Late labs will not be accepted.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Course Webpage
We will be using Canvas to post information relating to thecourse, including slides from the lectures.
We will be coordinating assignments and exams with theother sections of Math 316.
All sections of Math 316 will be using the Section 1Canvas site.
If you don’t have access to the Section 1 Canvas site,please let me know so I can add you.
The Section 3 Canvas site will not be used.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Di↵erential Equation
A di↵erential equation is a mathematical equation thatrelates some unknown function with its derivatives.
Example
Which equations are di↵erential equations?
(a) y
3 + 2y 2 + y = sin t
(b)Ry dx = y + cos t
(c) u
t
= ku
xx
(d) y
n+1
= y
n
+�t · F (t)
(e) @2
u
@t2 = c
2
@2
u
@x2
(f) ydx + xdy = 0
(g) x
0 + x sin y = cos y
(h) (sin t)0 = cos t
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Di↵erential Equation
A di↵erential equation is a mathematical equation thatrelates some unknown function with its derivatives.
Example
Which equations are di↵erential equations?
(a) y
3 + 2y 2 + y = sin t
(b)Ry dx = y + cos t
(c) u
t
= ku
xx
(d) y
n+1
= y
n
+�t · F (t)
(e) @2
u
@t2 = c
2
@2
u
@x2
(f) ydx + xdy = 0
(g) x
0 + x sin y = cos y
(h) (sin t)0 = cos t
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Newton’s Law of Heating/Cooling
The rate of change of the temperature of an object isproportional to the di↵erence between its own temperatureand the temperature of its surroundings.
u = Temperature of the co↵eeT = Room Temperaturek = Transmission coe�cient
du
dt
= ±k(u � T )
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Newton’s Law of Heating/Cooling
The rate of change of the temperature of an object isproportional to the di↵erence between its own temperatureand the temperature of its surroundings.
u = Temperature of the co↵eeT = Room Temperaturek = Transmission coe�cient
du
dt
= �k(u � T )
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
What are the independent variables, dependent variables, andparameters for the DE?
u
0 = �k(u � T )
t is an independent variableu is a dependent variablek and T are parameters
Alternative, but equivalent, forms:
u
0 = �k(u � T ),du
dt
= �k(u � T ), u = �k(u � T )
Note: The dot notation is only used for derivatives withrespect to time.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
What are the independent variables, dependent variables, andparameters for the DE?
u
0 = �k(u � T )
t is an independent variableu is a dependent variablek and T are parameters
Alternative, but equivalent, forms:
u
0 = �k(u � T ),du
dt
= �k(u � T ), u = �k(u � T )
Note: The dot notation is only used for derivatives withrespect to time.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
What are the independent variables, dependent variables, andparameters for the DE?
u
0 = �k(u � T )
t is an independent variableu is a dependent variablek and T are parameters
Alternative, but equivalent, forms:
u
0 = �k(u � T ),du
dt
= �k(u � T ), u = �k(u � T )
Note: The dot notation is only used for derivatives withrespect to time.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
What are the independent variables, dependent variables, andparameters for the DE?
u
0 = �k(u � T )
t is an independent variableu is a dependent variablek and T are parameters
Alternative, but equivalent, forms:
u
0 = �k(u � T ),du
dt
= �k(u � T ), u = �k(u � T )
Note: The dot notation is only used for derivatives withrespect to time.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
What are the independent variables, dependent variables, andparameters for the DE?
u
0 = �k(u � T )
t is an independent variableu is a dependent variablek and T are parameters
Alternative, but equivalent, forms:
u
0 = �k(u � T ),du
dt
= �k(u � T ), u = �k(u � T )
Note: The dot notation is only used for derivatives withrespect to time.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Solution of a DE
A function is a solution of a DE if, when we plug it (and itsvarious derivatives) into the DE, the equation is satisfied.
Example
Is u = T + e
�kt a solution to u
0 = �k(u � T )?
The derivative is u0 = �e
�kt . Plugging u and u
0 into the DEgives
�ke
�kt = �k(T + e
�kt
� T )
�ke
�kt = �ke
�kt
Yes!
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Solution of a DE
A function is a solution of a DE if, when we plug it (and itsvarious derivatives) into the DE, the equation is satisfied.
Example
Is u = T + e
�kt a solution to u
0 = �k(u � T )?
The derivative is u0 = �e
�kt . Plugging u and u
0 into the DEgives
�ke
�kt = �k(T + e
�kt
� T )
�ke
�kt = �ke
�kt
Yes!
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: General Solution of a DE
The general solution of a DE is a family of functions thatcontains all possible solutions to the DE.
Definition: Particular Solution of a DE
A particular solution of a DE is a solution that contains noarbitrary constants.
Example
The general solution to u
0 = �k(u � T ) is u = T + ce
�kt .Find a particular solution.
Set c equal to any number. For example, u = T + 7e�kt .
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: General Solution of a DE
The general solution of a DE is a family of functions thatcontains all possible solutions to the DE.
Definition: Particular Solution of a DE
A particular solution of a DE is a solution that contains noarbitrary constants.
Example
The general solution to u
0 = �k(u � T ) is u = T + ce
�kt .Find a particular solution.
Set c equal to any number. For example, u = T + 7e�kt .
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Equilibrium Solution of a DE
An equilibrium solution of a DE is a constant solution.Equilibrium solutions are sometimes referred to as criticalpoints, fixed points, or stationary points.
Example
Find an equilibrium solution to u
0 = �k(u � T ).
Let u = K , a constant. If we plug in K for u, we get
0 = �k(K � T ) =) K = T (assuming k 6= 0),
so y = T is an equilibrium solution.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Equilibrium Solution of a DE
An equilibrium solution of a DE is a constant solution.Equilibrium solutions are sometimes referred to as criticalpoints, fixed points, or stationary points.
Example
Find an equilibrium solution to u
0 = �k(u � T ).
Let u = K , a constant. If we plug in K for u, we get
0 = �k(K � T ) =) K = T (assuming k 6= 0),
so y = T is an equilibrium solution.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: (First Order) Autonomous DE
A first order autonomous DE is an equation of the form
dy
dx
= f (y)
Example
Which DE’s are autonomous?
(a) y
0 = �y + t
(b) x
0 = sin x
(c) y
0 =p
k
2/y � 1
(d) u
0 + ku = kT
0
+ kA sin!t
(e) p
0 = rp(1� p/K )
(f) x
0 = sin (tx)
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: (First Order) Autonomous DE
A first order autonomous DE is an equation of the form
dy
dx
= f (y)
Example
Which DE’s are autonomous?
(a) y
0 = �y + t
(b) x
0 = sin x
(c) y
0 =p
k
2/y � 1
(d) u
0 + ku = kT
0
+ kA sin!t
(e) p
0 = rp(1� p/K )
(f) x
0 = sin (tx)
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Theorem: Linearization About an Equilibrium Point.
Let y1
be an equilibrium point of a DE of the form y
0 = f (y).Assume that f has continuous derivative in a vicinity of y
1
.
(i) If f 0(y1
) < 0, then y
1
is an asymptotically stableequilibrium point.
(ii) If f 0(y1
) > 0, then y
1
is an unstable equilibrium point.
(iii) If f 0(y1
) = 0, then more information is needed to classifyy
1
.
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Example
Sketch the phase line for
y
0 = �k(y � 1)2, k > 0
and classify the stability of each critical point.(Refer to the chalkboard.)
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
The standard form for a first order di↵erential equation is
dy
dt
= f (t, y).
If t appears explicitly in the expression for f , then the DE issaid to be nonautonomous.
Can nonautonomous DE’s be analyzed using a phase line?
No
Instead, we use a direction field (also known as a slope field).
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
The standard form for a first order di↵erential equation is
dy
dt
= f (t, y).
If t appears explicitly in the expression for f , then the DE issaid to be nonautonomous.
Can nonautonomous DE’s be analyzed using a phase line?
No
Instead, we use a direction field (also known as a slope field).
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Example
Plot the slope field and a few solution curves for
y
0 = 1� y � sin (2t).
y ", t !
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Order of a DE
The order of a DE is the order of the highest derivative,ordinary or partial, that appears in the equation.
Example
What is the order of each DE?
(a) u
t
= ku
xx
(b) ydx + xdy = 0
(c) @2
u
@t2 = c
2
@2
u
@x2
(d) x
0x
(4) + x sin y = cos y
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Order of a DE
The order of a DE is the order of the highest derivative,ordinary or partial, that appears in the equation.
Example
What is the order of each DE?
(a) u
t
= ku
xx
2
(b) ydx + xdy = 0 1
(c) @2
u
@t2 = c
2
@2
u
@x2 2
(d) x
0x
(4)+ x sin y = cos y 4
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Linear DE
An nth order ordinary di↵erential equation (ODE) is said to belinear if it can be written in the form
a
0
(t)y (n) + a
1
(t)y (n�1) + · · ·+ a
n
(t)y = g(t).
The functions a0
, a1
, . . . , an
are called the coe�cients of theequation. An ODE that is not linear is called nonlinear.
Example
Which ODE’s are linear?
(a) (1 + y)d2
y
dt
2
+ t
dy
dt
+ y = e
t
(b) d
2
y
dt
2
+ sin (t + y) = sin t
(c) d
3
y
dt
3
+ t + (cos2 t)y = t
3
(d) dy
dt
+ ty
2 = 0
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Linear DE
An nth order ordinary di↵erential equation (ODE) is said to belinear if it can be written in the form
a
0
(t)y (n) + a
1
(t)y (n�1) + · · ·+ a
n
(t)y = g(t).
The functions a0
, a1
, . . . , an
are called the coe�cients of theequation. An ODE that is not linear is called nonlinear.
Example
Which ODE’s are linear?
(a) (1 + y)d2
y
dt
2
+ t
dy
dt
+ y = e
t
(b) d
2
y
dt
2
+ sin (t + y) = sin t
(c) d
3
y
dt
3
+ t + (cos2 t)y = t
3
(d) dy
dt
+ ty
2 = 0
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Homogeneous DE
A linear ODE of the form
a
0
(t)y (n) + a
1
(t)y (n�1) + · · ·+ a
n
(t)y = g(t)
is called homogeneous if g(t) = 0 for all t. Otherwise, theequation is nonhomogeneous.
Example
Which DE’s are homogeneous?
(a) d
2
y
dt
2
= ty
(b) dQ
dt
= �
�1
1+t
�Q + 2 sin t
(c) d
dx
⇥p(x)dy
dx
⇤= r(x)y
(d) y
0 + sin t = y
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Homogeneous DE
A linear ODE of the form
a
0
(t)y (n) + a
1
(t)y (n�1) + · · ·+ a
n
(t)y = g(t)
is called homogeneous if g(t) = 0 for all t. Otherwise, theequation is nonhomogeneous.
Example
Which DE’s are homogeneous?
(a) d
2
y
dt
2
= ty
(b) dQ
dt
= �
�1
1+t
�Q + 2 sin t
(c) d
dx
⇥p(x)dy
dx
⇤= r(x)y
(d) y
0 + sin t = y
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Definition: Initial Value Problem (IVP)
An initial value problem is a DE
y
0 = f (t, y)
along with a point (t0
, y0
) in the domain of f called the initial
condition.
Example
A cup of co↵ee has a temperature of 200�F when freshlypoured and is left in a room at 70�F. One minute later, theco↵ee has cooled to 190�F.
(a) Write an IVP that models the temperature of the co↵ee.
(b) How long will it take for the co↵ee to reach 170�F?
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
(a). Newton’s law of heating and cooling says that
u
0 = �k(u � T )
We know that T = 70�F and the initial temperature of theco↵ee is 200�F, so the IVP is
u
0 = �k(u � 70), u(0) = 200�F
(b). We know that the general solution to the DE isu = T + ce
�kt . We can find c using the initial condition.
200 = 70 + ce
0 =) c = 130
We can find k since we know the temperature after 1 minute.
190 = 70 + 130e�k·1 =) k = ln (13/12)
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
(b). To find the time when the temperature reaches 170�F, letu = 170 and solve for t.
170 = 70 + 130e�t ln (13/12)
170 = 70 + 130
✓12
13
◆t
ln(10/13) = t ln(12/13)
t =ln(10/13)
ln(12/13)⇡ 3.278 min
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Example
Plot the phase line (one-dimensional phase portrait) for
u
0 = �k(u � T ), k > 0.
(Refer to the chalkboard.)
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
Example
The population of rabbits in a field is given by
y = ry
⇣1�
y
K
⌘, r ,K > 0,
where r is the growth rate and K is the carrying capacity.Sketch the phase line and determine the stability of eachcritical point. (Refer to the chalkboard.)
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations
(Gary Marple) January 9th, 2019 Math 316: Di↵erential Equations