Differential Equations Answers

Consider and solve the initial value problem of a new form of life discovered on a distant

planet. Outside its habitable zone, the rate of change of population of the life form is

governed by the following data:

dy

dx + y = f(x), where f(x) = {

e

−x

, 0 ≤ x < 2

e

x

, x ≥ 2

; y(0) = 1

Consider a flask that contain 3 liters of salt water. Suppose that water containing 25 grams

per liters of salt is pumped into the flask at the rate of 2 liters per hour, and the mixture,

being steadily stirred, is pumped out of the flask at the same rate. Find a differential

equation satisfied by the amount of salt f(t) in the flask at time t.

A pond on a fish farm has a carrying capacity of 1000 fish. The pond was originally stocked

with 100 fish. Let N(t) denote the number of fish in the pond after t months.

a) Set up a logistic differential equation satisfied by N(t), and plot an approximate graph

of a fish population.

b) Find the size of the population of fish with the highest rate of growth. Find this rate

given that the intrinsic rate of growth is 3.

Using X = x - 2,Y = y +1, reduce the equation 4(x-2)^2 dx/dy =(x + y - 2)^2 to the homogeneous form of 1st order equation.

using the lagrange’s method, solve the differential equation (xy^3 - 2x^4)b + (2y^4 - x^3y)q = 9z(x^3- y^3)

(px-y)(py+x)=a^2

The rate at which a super computer body cools is proportional to the difference between the temperature of the body and that of the surrounding air. If a body in air at will cool from to in one minute, find its temperature at the end of three minutes.