Agendagmarple/Lec_01.pdf · Agenda Introduction Sections 1.1-1.3 Reminders Read §1.1-1.3 & 2.1 Do...

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.


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.


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.


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.


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.


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.


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


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


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 )


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 )


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.



u

0 = �k(u � T )



u

0 = �k(u � T ),du

dt

= �k(u � T ), u = �k(u � T )




u

0 = �k(u � T )



u

0 = �k(u � T ),du

dt

= �k(u � T ), u = �k(u � T )




u

0 = �k(u � T )



u

0 = �k(u � T ),du

dt

= �k(u � T ), u = �k(u � T )




u

0 = �k(u � T )



u

0 = �k(u � T ),du

dt

= �k(u � T ), u = �k(u � T )



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!


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!


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 .


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 .


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.


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.


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)


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)


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

.


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.)


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).


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).


Example

Plot the slope field and a few solution curves for

y

0 = 1� y � sin (2t).

y ", t !


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


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


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


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


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


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


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?


(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)


(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


Example

Plot the phase line (one-dimensional phase portrait) for

u

0 = �k(u � T ), k > 0.

(Refer to the chalkboard.)


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.)



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Agendagmarple/Lec_01.pdf · Agenda Introduction Sections 1.1-1.3 Reminders Read §1.1-1.3 & 2.1 Do...

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