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
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.
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.
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