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.
EXERCISE: DIFFERENTIAL EQUATIONS
10.
b)
Find the temperature of the chicken 10 minutes after it is removed from the oven.
c)
How long will it take for the chicken to cool off to a room temperature of 70 o F.
Given a differential equation as
dv k 0 for which k is a constant. dt t 3
a)
Find the equation of v in terms of k and t if v 1 3 .
b)
What will be the equation of v in terms of t if v 5 when t approaches ?
ANSWERS: 3
1.
General solution: y Ae
(3 x 2 3) 2
c 3
(3 x 2 3) 2
Particular solution: y e e t ce t
2.
v
3.
y x 2 ln x x 2 c x
5.
a)
y sin x c (cos x)
b)
y 2ecos x
3
3
3
6.
y
7.
y
8.
y
9.
a)
T 230e0.1902t
b)
T 104.33 F
c)
40 minutes
a)
v 3
k k 2 2 2t
b)
v 5
2
10.
e
3 x
2
1 60 ( x 1)2 4 x 1 2 e
cos x
c
t2