代做AMATH 351 Introduction to Differential Equations and Applications Homework 1代写Python语言

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Homework 1

AMATH 351

Introduction to Differential Equations and Applications

Due: Wednesday, October 9 at 11:59pm on Gradescope

Directions:

Complete all component skills exercises as neatly as possible. Up to 2 points may be deducted for homework that is illegible and/or poorly organized. You are encouraged to type homework solutions, and a half bonus point will be awarded to students who use LATEX, provided the homework is complete. (Check out the LATEX beginner document and overleaf.com if you are new to LATEX.) If you prefer not to type homeworks,I ask that homeworks be scanned. (I will not accept physical copies.) In addition, homeworks must be in .pdf format. If you are handwriting the homework, you may attach the image files separately in Gradescope.

Direction fields and phase line plots

These problems are designed to allow you practice drawing and interpreting direction fields (C.S. 1.3) and phase line plots (C.S. 1.4). You must use either MATLAB or Mathematica (or python) to plot these. Please make them look as nice and as readable as you can.

• Use Mathematica. The software is available through the University here: https://phys.washington.edu/mathematica.

See the example notebook (direction fields and phase planes.nb) included on the course modules page.

• Use MATLAB. The software is available through the University here:

https://itconnect.uw.edu/wares/uware/matlab/

See the example file, direction fields and phase planes .m, included on the course modules page.

Include the figures in your submission (you may attach them separately.) You may find this online tool for combining pdf files useful: https://combinepdf.com/.

1. Recall from the lectures that we can use direction fields to describe the ultimate behavior. of solutions of first- order ODEs without actually solving them. Sketch the direction fields (see note below) for the following ODEs and answer the subsequent questions.

(a) Consider y′ = xcos(y). What are the equilibrium solutions? Draw the direction field between 0 ≤ x ≤ 2π and −π ≤ y ≤ π . Plot with equally sized vectors, i.e., if using Mathematica, use the qualifier ‘VectorScale − > {Small, Small, None}’). Given that y(0) = 0, what value does y(x) approach as x → ∞? What if y(0) = π? What is the stability of the two equilibrium points in the range (−π,π)?

(b) Consider y′ = xy with y(0) = 1. What are the equilibrium solutions? Draw the direction field between −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2. Plot with equally sized vectors and 11 vectors in both directions, i.e., if using Mathematica, use the qualifier ‘VectorScale − > {Small, Small, None}, VectorPoints − > 11’). What is the limiting value of y(x) when (i) x → ∞, (ii) x → −∞? What about if y(0) = −1? What can you say about the stability of the equilibrium point(s)?

2. For the following equations draw a phase line plot then determine the equilibrium points and classify each one as stable, unstable, or semi-stable.

(a) y′ = 4y − y3 , −∞ < y0 < ∞ (b) y′ = ey − 1, −∞ < y0 < ∞

3. Thus far we have only used phase line plots to analyze differential equations of the form. y′ = f(y). For this class of first order ODEs, phase line plots have allowed us to find and classify equilibrium solutions that are constant functions (i.e. they are of the form. y(x) = c for some constant c). However, this methodology can be extended to differential equations of the form. y′ = f(x,y) for simple enough functions f(x,y). Consider the differential equation

y′ = x2 − y.

Treating x as a constant, find the equilibrium solution of this ODE. Is it stable, unstable, or semi-stable? Now,

allowing x to vary moves around the location of the equilibrium solution, but doesn’t change its stability. What

do you think happens to solutions y for large values of x (if you’re unsure, feel free to plot the direction field)?

Now consider a similar ODE,

y′ = y − x2 .

Again, treating x as a constant, find and classify the equilibrium solution. Allowing x to vary again moves around the location of the equilibrium solution without affecting its stability. What do you think happens to solutions y for large values of x? Note: your answer in this case will depend on whether or not the solution lies above or below the equilibrium solution.

Separable equations

The following problems are designed to help you become comfortable with solving separable equations (C.S. 2.1). They will also allow you to classify differential equations (C.S. 1.2) and practice checking solutions to differential equations (C.S. 1.1).

4. First, classify the following by order, dimension, as linear or nonlinear, and as autonomous or nonautonomous. If linear, is it constant coefficient or variable coefficient and is it homogeneous or nonhomoegeneous.

Then, solve the differential equation (or initial value problem) using the method of separation. If no initial value is given, give the general form. of the solution. Unless otherwise stated, find the explicit form. of the solution. If you find an explicit solution solution, verify that it is indeed a solution by checking whether it satisfies the differential equation or initial value problem.

(a) y ′ = 1 + 3y 2/1 + sin(t). Leave the solution in implicit form.

(b) y′ = −(3t2 + 1)y2 , y(1) = 1

5. First, classify the following by order, dimension, as linear or nonlinear, and as autonomous or nonautonomous. If linear, is it constant coefficient or variable coefficient and is it homogeneous or nonhomoegeneous.

Then solve the differential equation (or initial value problem) using the method of separation. If no initial value is given, give the general form. of the solution. Unless otherwise stated, find the explicit form. of the solution.

(a) dx/dy = 4/3πe3x (4 + y2), y(0) = 2

(b) y′ = xex csc(y),y(0) = 3/π. Leave the solution in implicit form.