Math 3B: Lecture 23 Noah White November 16, 2016 Announcements - - PowerPoint PPT Presentation

▶
math 3b lecture 23
SMART_READER_LITE
LIVE PREVIEW

Math 3B: Lecture 23 Noah White November 16, 2016 Announcements - - PowerPoint PPT Presentation

Apr 14, 2023 165 likes •530 views

Math 3B: Lecture 23 Noah White November 16, 2016 Announcements Homework is due this Friday at 2pm Announcements Homework is due this Friday at 2pm Review lecture Friday, see Piazza for vote on topics. Announcements Homework is

slide-1
SLIDE 1

Math 3B: Lecture 23

Noah White November 16, 2016

slide-2
SLIDE 2

Announcements

  • Homework is due this Friday at 2pm
slide-3
SLIDE 3

Announcements

  • Homework is due this Friday at 2pm
  • Review lecture Friday, see Piazza for vote on topics.
slide-4
SLIDE 4

Announcements

  • Homework is due this Friday at 2pm
  • Review lecture Friday, see Piazza for vote on topics.
  • Lectures 11/28 and 11/30 change. Vote on Piazza
slide-5
SLIDE 5

Slope fields

We want to study differential equations of the form dy dt = f (t, y) These could be quite complicated. Most of the time, they are not solvable!

slide-6
SLIDE 6

Slope fields

We want to study differential equations of the form dy dt = f (t, y) These could be quite complicated. Most of the time, they are not solvable!

Aim

Get a qualitative understanding for how a solution behaves, given an initial condition y(t0) = y0.

slide-7
SLIDE 7

Slope fields

We want to study differential equations of the form dy dt = f (t, y) These could be quite complicated. Most of the time, they are not solvable!

Aim

Get a qualitative understanding for how a solution behaves, given an initial condition y(t0) = y0.

Key tool

Slope fields. At every point on the yt-plane we draw a small line segment (a vector) with slope f (y, t).

slide-8
SLIDE 8

Examples

Note

If we want to draw a slope field, we cannot actually draw a line segment for every point. Instead we pick a grid of points in the plane.

slide-9
SLIDE 9

Examples

Note

If we want to draw a slope field, we cannot actually draw a line segment for every point. Instead we pick a grid of points in the plane.

Examples

Lets use Geogebra! Here is the command we will use: SlopeField[f(x,y)] will produce a slope field for the equation dy dx = f (x, y)

slide-10
SLIDE 10

Sketching solutions

Using the slope field we can sketch rough pictures of the solution, given a starting point (an initial condition).

Note

These pictures are not supposed to be perfect. But they will hopefully give you an idea of

slide-11
SLIDE 11

Sketching solutions

Using the slope field we can sketch rough pictures of the solution, given a starting point (an initial condition).

Note

These pictures are not supposed to be perfect. But they will hopefully give you an idea of

  • when doees the solution increase/decrease?
slide-12
SLIDE 12

Sketching solutions

Using the slope field we can sketch rough pictures of the solution, given a starting point (an initial condition).

Note

These pictures are not supposed to be perfect. But they will hopefully give you an idea of

  • when doees the solution increase/decrease?
  • what does the solution do in the long term?
slide-13
SLIDE 13

Sketching solutions

Using the slope field we can sketch rough pictures of the solution, given a starting point (an initial condition).

Note

These pictures are not supposed to be perfect. But they will hopefully give you an idea of

  • when doees the solution increase/decrease?
  • what does the solution do in the long term?
  • is the solution ever above to below a certain value?
slide-14
SLIDE 14

Sketching solutions

Using the slope field we can sketch rough pictures of the solution, given a starting point (an initial condition).

Note

These pictures are not supposed to be perfect. But they will hopefully give you an idea of

  • when doees the solution increase/decrease?
  • what does the solution do in the long term?
  • is the solution ever above to below a certain value?

Examples

Lets use Geogebra again.

slide-15
SLIDE 15

Nullclines

Definition

The nullcline for dy

dt = f (t, y) is the set of points (t, y) where

f (t, y) = 0

slide-16
SLIDE 16

Nullclines

Definition

The nullcline for dy

dt = f (t, y) is the set of points (t, y) where

f (t, y) = 0

Examples

Lets use Geogebra!

slide-17
SLIDE 17

Drawing slope fields by hand

Drawing slope fields by hand can be difficult! But we can use the nullclines to get an approximate picture

Examples

Lets draw some on the board.

slide-18
SLIDE 18

Autonomous equations

Deafinition

An ODE of the form dy dt = f (y) i.e. where the right hand side does not depend on t, is called autonomous

slide-19
SLIDE 19

Autonomous equations

Deafinition

An ODE of the form dy dt = f (y) i.e. where the right hand side does not depend on t, is called autonomous

Important property

The nullclines of an autonomous equation are horizontal straight lines! Nullclines = equilibrium solutions

slide-20
SLIDE 20

Autonomous equations

Deafinition

An ODE of the form dy dt = f (y) i.e. where the right hand side does not depend on t, is called autonomous

Important property

The nullclines of an autonomous equation are horizontal straight lines! Nullclines = equilibrium solutions

slide-21
SLIDE 21

Autonomous equations

Deafinition

An ODE of the form dy dt = f (y) i.e. where the right hand side does not depend on t, is called autonomous

Important property

The nullclines of an autonomous equation are horizontal straight lines! Nullclines = equilibrium solutions We want points (t, y) such that f (y) = 0.

  • Suppose f (a) = 0.
slide-22
SLIDE 22

Autonomous equations

Deafinition

An ODE of the form dy dt = f (y) i.e. where the right hand side does not depend on t, is called autonomous

Important property

The nullclines of an autonomous equation are horizontal straight lines! Nullclines = equilibrium solutions We want points (t, y) such that f (y) = 0.

  • Suppose f (a) = 0.
  • Then (t, a) is on the nullcline, for any t.
slide-23
SLIDE 23

Autonomous equations

Deafinition

An ODE of the form dy dt = f (y) i.e. where the right hand side does not depend on t, is called autonomous

Important property

The nullclines of an autonomous equation are horizontal straight lines! Nullclines = equilibrium solutions We want points (t, y) such that f (y) = 0.

  • Suppose f (a) = 0.
  • Then (t, a) is on the nullcline, for any t.
  • So the line y = a is part of the nullcline, whenever f (a) = 0.
slide-24
SLIDE 24

Slope fields and nullclines for autonomous systems

Thus our slope field and nullclines look something like

slide-25
SLIDE 25

Phase lines/diagram

Thus our slope field and nullclines look something like

slide-26
SLIDE 26

Phase lines/diagram

Thus our slope field and nullclines look something like

slide-27
SLIDE 27

Phase lines/diagram

Thus our slope field and nullclines look something like unstable stable

slide-28
SLIDE 28

Phase lines

Recipe to draw phase lines

slide-29
SLIDE 29

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
slide-30
SLIDE 30

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
  • 2. Draw dots where equilibrim solutions live
slide-31
SLIDE 31

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
  • 2. Draw dots where equilibrim solutions live
  • 3. Draw up arrows on intervals between dots where the derivative

is positive

slide-32
SLIDE 32

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
  • 2. Draw dots where equilibrim solutions live
  • 3. Draw up arrows on intervals between dots where the derivative

is positive

  • 4. Draw down arrows on intervals between dots where the

dericative is negative

slide-33
SLIDE 33

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
  • 2. Draw dots where equilibrim solutions live
  • 3. Draw up arrows on intervals between dots where the derivative

is positive

  • 4. Draw down arrows on intervals between dots where the

dericative is negative

Definition

slide-34
SLIDE 34

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
  • 2. Draw dots where equilibrim solutions live
  • 3. Draw up arrows on intervals between dots where the derivative

is positive

  • 4. Draw down arrows on intervals between dots where the

dericative is negative

Definition

  • An equalibrium is stable if the two arrows are pointing towards

it.

slide-35
SLIDE 35

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
  • 2. Draw dots where equilibrim solutions live
  • 3. Draw up arrows on intervals between dots where the derivative

is positive

  • 4. Draw down arrows on intervals between dots where the

dericative is negative

Definition

  • An equalibrium is stable if the two arrows are pointing towards

it.

  • It is unstable if the two arrows are pointing away from it.
slide-36
SLIDE 36

Phase lines

Recipe to draw phase lines

  • 1. Draw a vertical corresponding to y axis
  • 2. Draw dots where equilibrim solutions live
  • 3. Draw up arrows on intervals between dots where the derivative

is positive

  • 4. Draw down arrows on intervals between dots where the

dericative is negative

Definition

  • An equalibrium is stable if the two arrows are pointing towards

it.

  • It is unstable if the two arrows are pointing away from it.
  • It is semistable if the arrows point in the same direction.