Differential Equations

Page 1

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 2e3 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  13 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.


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