辅导MAT244H1S、Java编程语言讲解、辅导c/c++，Python
- 首页 >> Java编程 MAT244H1S FINAL ASSIGNMENT
DUE TUESDAY APRIL 14 AT 9AM
Open book, open notes
Exam policies:
• You may discuss the problems with your classmates but must write up your solutions
independently. Credit will be divided equally between identical submissions.
• Solve each problem on its own page (or set of pages). After you are done, upload your work
for each problem as a pdf to the corresponding problem on Crowdmark.
• You may write your solutions on paper or, if you prefer, on a computer. Should you write
on paper, please scan or take a high quality picture of each page, and combine them into one
file before uploading. The Office Lens application for Android or iOS can combine multiple
scanned pages into a single PDF file (but you may use any tool that works for you).
• Detail the main steps of your solution (e.g. if you need to compute the eigenvalues of
a matrix A, you should write down its characteristic polynomial). Your reasoning will
be at least as important in our scoring as your final numerical answer. You may receive
substantial partial credit if your approach is sound but you make an arithmetic error at the
end.
• You may refer to the textbook, any notes from lecture, and any notes posted on Quercus.
1
2 DUE TUESDAY APRIL 14 AT 9AM
1. (10 points) Solve each of the following initial value problems, and express the solution as a
explicit function of t.
2. (10 points) For real parameters b and c, consider the equation.
Describe, both in terms of inequalities and with a sketch, the region of the bc plane such
that every solution satisfies
limt→∞y(t) = 0.
3. (10 points) Find the general solution to the equation
y
(4) − 2y
00 + y = cost.
4. (20 points) Consider the system
dxdt = x(1 − x − y),dydt = y(2 − x − y).
First determine the critical points. Then for each critical point:
(a) Find the corresponding linear system near the critical point.
(b) Determine the stability and type of the critical point, and sketch some nearby trajectories.
5. (10 points) Let A be a real 4 × 4 matrix and consider the initial value problem
(1) = Ax, x(0) = ξ0,
where ξ0 ∈ R4 be a vector such that (A − 3I)
3
ξ0 = 0 but ξ1 := (A − 3I)ξ0 6= 0 and
ξ2 := (A − 3I)
2
ξ0 6= 0.
(a) Make the change of variable x = e
3ty, and derive an equation y
0 = By (find B).
(b) Solve the initial value problem
y
0 = By, y(0) = ξ0,
representing y in terms of ξ0, ξ1, and ξ2. [Suggestion: use the matrix exponential.]
(c) Solve the original initial value problem (1) in terms of ξ0, ξ1, and ξ2.
DUE TUESDAY APRIL 14 AT 9AM
Open book, open notes
Exam policies:
• You may discuss the problems with your classmates but must write up your solutions
independently. Credit will be divided equally between identical submissions.
• Solve each problem on its own page (or set of pages). After you are done, upload your work
for each problem as a pdf to the corresponding problem on Crowdmark.
• You may write your solutions on paper or, if you prefer, on a computer. Should you write
on paper, please scan or take a high quality picture of each page, and combine them into one
file before uploading. The Office Lens application for Android or iOS can combine multiple
scanned pages into a single PDF file (but you may use any tool that works for you).
• Detail the main steps of your solution (e.g. if you need to compute the eigenvalues of
a matrix A, you should write down its characteristic polynomial). Your reasoning will
be at least as important in our scoring as your final numerical answer. You may receive
substantial partial credit if your approach is sound but you make an arithmetic error at the
end.
• You may refer to the textbook, any notes from lecture, and any notes posted on Quercus.
1
2 DUE TUESDAY APRIL 14 AT 9AM
1. (10 points) Solve each of the following initial value problems, and express the solution as a
explicit function of t.
2. (10 points) For real parameters b and c, consider the equation.
Describe, both in terms of inequalities and with a sketch, the region of the bc plane such
that every solution satisfies
limt→∞y(t) = 0.
3. (10 points) Find the general solution to the equation
y
(4) − 2y
00 + y = cost.
4. (20 points) Consider the system
dxdt = x(1 − x − y),dydt = y(2 − x − y).
First determine the critical points. Then for each critical point:
(a) Find the corresponding linear system near the critical point.
(b) Determine the stability and type of the critical point, and sketch some nearby trajectories.
5. (10 points) Let A be a real 4 × 4 matrix and consider the initial value problem
(1) = Ax, x(0) = ξ0,
where ξ0 ∈ R4 be a vector such that (A − 3I)
3
ξ0 = 0 but ξ1 := (A − 3I)ξ0 6= 0 and
ξ2 := (A − 3I)
2
ξ0 6= 0.
(a) Make the change of variable x = e
3ty, and derive an equation y
0 = By (find B).
(b) Solve the initial value problem
y
0 = By, y(0) = ξ0,
representing y in terms of ξ0, ξ1, and ξ2. [Suggestion: use the matrix exponential.]
(c) Solve the original initial value problem (1) in terms of ξ0, ξ1, and ξ2.