Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17,...
Transcript of Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17,...
![Page 1: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/1.jpg)
1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd.
Homework:
Day 5: July 6thChapter: 1.5 Existence and Uniqueness.
Chapter: 1.6 Equilibria and Phase Line.
![Page 2: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/2.jpg)
Coming up
Midterm 1 on Friday, July 9th: Chapter 1
Lab 1 due Thursday next week,July 15th
![Page 3: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/3.jpg)
Existence and Uniquenessdy
dt= F (y, t)
Then: There exists a unique solution to initial value problem y(t0) = y0
defined for t0 !A < t < t0 + A
If: Continuously differentiable in and , i.e.ty
is differentiable in and and the derivative is continuousF y t
![Page 4: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/4.jpg)
Existence and Uniquenessdy
dt= F (y, t)
Then: There exists a unique solution to initial value problem y(t0) = y0
defined for t0 !A < t < t0 + A
If: Continuously differentiable in and , i.e.ty
Result: Solutions can not cross.
Example: Homework. Chapter 1.5: 1,3,5 & 7.
![Page 5: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/5.jpg)
Example: Homework. Chapter 1.5: 1,3,5 & 7.
1. dy
dt= f(t, y)
3. dy
dt= f(t, y)
y1(t) = 3 for all t is a solution
for all t is a solution
for all t is a solution
y1(t) = t + 2
what can we say about y(0) = 1
what can we say about y(0) = 1y2(t) = !t2
![Page 6: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/6.jpg)
Example: Homework. Chapter 1.5: 1,3,5 & 7.
5.
7.
dy
dt= (y ! 2)(y ! 3)y
what can we say about
what can we say about y(0) = 1
y(0) = 4
![Page 7: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/7.jpg)
E & U: Examples
dy
dt=
y
t
dy
dt=
y
t2
dy
dt=
32y
13
When existence or uniqueness fails
Look for “bad” points
1.
2.
3.
![Page 8: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/8.jpg)
E & U: Examplesdy
dt=
y
t
Lots of solutions satisfy: y(0) = 0
but none satisfy y(0) = k != 0
1.
![Page 9: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/9.jpg)
E & U: Examplesdy
dt=
y
t22. Homework: 1.5 #13
y(t) = 0 is an equilibrium solution
![Page 10: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/10.jpg)
E & U: Examplesdy
dt=
y
t2
y(t) = De!1t
y(t)! 0t! 0+,As
t! 0!, y(t)! ±"
2. Homework: 1.5 #13
y(t) = 0 is an equilibrium solution
![Page 11: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/11.jpg)
E & U: Examples
3. derivative not continuous at 0
y(t) = 0 is an equilibrium solution
dy
dt=
32y
13
Lots of solutions satisfy: y(0) = 0
![Page 12: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/12.jpg)
Phase LineAutonomous equations
dy
dt= F (y)
slopes do not depend on t
sign of shows whether solutions increase or decrease.
F
![Page 13: Day 5: July 6thmath.bu.edu/people/baldur/MA226day5.pdf · 1.6 Page 91: #1, 3, 5, 7, 13, 15, 17, 29-35 odd. Homework: Day 5: July 6th Chapter: 1.5 Existence and Uniqueness. Chapter:](https://reader034.fdocuments.us/reader034/viewer/2022052011/60278256ab58a51cb4161ee8/html5/thumbnails/13.jpg)
Ex: Logistic equation
dy
dt= y(1! y)