代做Mathematics 376 Ordinary Differential Equations Assignment 1代做Python编程

Mathematics 376: Ordinary Differential Equations

Assignment 1

Note: This assignment consists of 10 problems of equal weight.

Due: After Unit 6

1. Solve the following initial value problem,

2. Find a special integrating factor and solve

3. Find an integrating factor and solve

4. Solve the following initial value problem,

5. Solve

6. Solve

7. Solve

8. A tank is filled with V = 200 L of a brine containing α = .4 kg of salt A per litre. At moment 0, input and output valves are opened, and a brine containing another salt B, with concentration β = .2 kg per litre runs into the tank at a rate ri = 5 L/sec. The mix runs out of the tank with rate ro = 4 L/sec. The salts do not interact with each other. Determine the ratio k of quantity of salt B to the quantity of salt A when the tank contains V1 = 400 L of the mixture.

9. (Heating)

The temperature M(t) outside a building decreases at a constant rate of 1 ◦C per hour. The inside of the building is heated, and there is no other source of cooling. The heater was switched on at time t = 0, when the temperature inside, T(t), was 17◦C, and the temperature outside was 0 ◦C. Assume that the heater generates a constant amount h = 50,000 Btu/hr of heat when it is working, the heat capacity of the building is γ = 1/5 degrees per thousand Btu, and the time constant for heat transfer between the outside and the inside of the building is τ = 2 hr. On the basis of Newton’s law of cooling,

find the upper value of the temperature in the building in the time interval 0 ≤ t < 4 hr.

10. (Landing)

A container with mass M kg is dropped by a helicopter from height H km at time t = 0, with zero velocity. From the outset, its fall is controlled by gravity and the force of air resistance, f(v) = −kv, where v is the current velocity of the container.

In τ seconds after the drop, a parachute opens, resulting in an increase of air resistance up to F(v) = −Kv. Determine the time T at which the container touches the ground, and its velocity at this moment, if

M = 200 kg, H = 2000 m, τ = 20 s, k = 10 kg/s, and K = 400 kg/s.