EXERCISE: DIFFERENTIAL EQUATIONS
1.
The gradient of a curve at a point (x, y) satisfies the differential equation dy 9xy 3x 2 2 dx
Obtain the general solution of the differential equation. Hence, find the equation of the curve that passes through the point (1, −e).
2.
A car moves from rest along a straight road. After t seconds, the velocity is v meters per dv second. The motion is modeled by v e t , where and are constants. Find dt v in terms of , and t. 3
3.
dy y 2x 2 ln x . The variables x and y of a curve satisfy the differential equation 2x dx
Find the general solution of the curve. 4.
A particle moves from rest in a straight line with velocity, v 4 4e 2t at any time t seconds. Show that the distance covered by the particle at time t = 2 is 6.0366 meters.
5.
a)
Solve the equation x
b)
Find the general solution for the differential equation
dy xy y if y(0) = 2. Give your answer y as a function of x. dx
dy y tan x sec x . Give your dx
answer y as a function of x.
dy y 2e3 x if y(0) = 1. Give your answer y as a function of x. dx
6.
Solve the equation
7.
Find the particular solution for the differential equation x 1
dy y x 13 if y = 1 dx
when x = 3. Give your answer y as a function of x. 8.
Find the particular solution for the differential equation
dy y 2 cos x e sin x 0 . dx 2
Give your answer y as a function of x if y 2 . 4
9.
Newton’s law of cooling states that the rate of at which the temperature T changes in a cooling body is proportional to the difference between the temperature in the body and the constant temperature surrounding the medium. When a chicken is removed from an oven, its temperature is measured at 300 Three minutes later its temperature is 200 o F. a)
Find an expression for the temperature of the chicken at time t.
o
F.