代写MAT244H1S、Java编程语言代做、代写c/c++，Python

日期：2020-04-15 10:24

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.