微分方程代写 MT3504代写 数学与统计代写 数学作业代写
MT3504: Differential Equations
Problem Sheet 1
微分方程代写 Section A: Problems. Attempt all questions. 1. Sketch the direction field for the following ODE and draw on some typical integral curves:
Section A: Problems. Attempt all questions. 微分方程代写
1. Sketch the direction field for the following ODE and draw on some typical integral curves:
Y’ = x + 2y.
Solve the ODE with the initial condition y(0) = 1 and again with the initial condition y(0) = −1. Then briefly compare your solutions with your sketch.
2. Solve the following first order, linear ODE using an integrating factor:
Xy’ + 2y = sin x, y(π/2) = 1.
Sketch your solution from x = π/2 to x = 4π. Discuss briefly what happens as x → ∞ and also the behaviour when 0 < x < π/2. 微分方程代写
![微分方程代写](https://www.writingessays.org/wp-content/uploads/2023/05/18-1024x215.png)
4. (Harder) Consider the nonlinear first order ODE
y = xy’ − (y’)2 .
Differentiate this equation with respect to x to find two possible solutions to the differential equation. Using the original ODE show that these solutions take the form:
y = ax − a2 and y = x2/4
where a is an arbitrary constant. Sketch the solutions for various values of a. What can you say about the relation between the two solution curves?
Note: The initial differential equation is a special case of Clairaut’s Equation, y = xy’ + g(y‘).
Section B: Examples. 微分方程代写
For practice. Attempt at least some parts, more if you finding a topic tricky.
![微分方程代写](https://www.writingessays.org/wp-content/uploads/2023/05/19.png)
6. Sketch a direction field for each of the following ODEs and draw a typical integral curve:
(i) y’ = y + 2
(ii) y’ = e−x + y
Note: To explore further than a simple hand drawn sketch you may wish to adapt the Python code provided for Examples 1.1 and 1.2 of the lecture notes.
![](https://www.writingessays.org/wp-content/uploads/2023/05/20.png)
![](https://www.writingessays.org/wp-content/uploads/2023/05/21.png)
![微分方程代写](https://www.writingessays.org/wp-content/uploads/2023/05/22.png)
10. Show that the following first order ODEs are homogeneous and find their solutions:
(i) xyy’ + y2 − x2 = 0
(ii) y’ = y/(y − x)
(iii) y’ = (2y − x)/(2x − y), y(0) = 1
![微分方程代写](https://www.writingessays.org/wp-content/uploads/2022/12/名-1024x439.jpg)
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