differential equation

446 DIFFERENTIAL EQUATIONS
Find the equation of the curve if it passes
through the point ( 1, 1 )
3 .
[
y = 3 ]
2 x2 − x3
6 − 1
7. The acceleration, a, of a body is equal to its
rate of change of velocity, dv . Find an equation
for v in terms of t, given that when t = 0,
dt
velocity v = u.
[v = u + at]
8. An object is thrown vertically upwards with
an initial velocity, u, of 20 m/s. The motion
of the object follows the differential equation
ds
= u−gt, where s is the height of the object
dt
in metres at time t seconds and g = 9.8m/s 2 .
Determine the height of the object after 3
seconds if s = 0 when t = 0. [15.9 m]
46.4 The solution of equations of the
form dy
dx = f ( y)
A differential equation of the form dy = f (y) is
dx
initially rearranged to give dx = dy and then the
f (y)
solution is obtained by direct integration,
i.e.
∫
∫
dx =
dy
f ( y)
Problem 6. Find the general solution of
dy
dx = 3 + 2y.
Rearranging dy = 3 + 2y gives:
dx
dx =
dy
3 + 2y
Integrating both sides gives:
∫
∫
dx =
dy
3 + 2y
Thus, by using the substitution u = (3 + 2y) — see
Chapter 39,
x =
2 1 ln (3 + 2y) + c (1)
It is possible to give the general solution of a differential
equation in a different form. For example, if
c = ln k, where k is a constant, then:
x =
2 1 ln(3 + 2y) + ln k,
i.e. x = ln(3 + 2y) 2 1 + ln k
or x = ln [k √ (3 + 2y)] (2)
by the laws of logarithms, from which,
e x = k √ (3 + 2y) (3)
Equations (1), (2) and (3) are all acceptable general
solutions of the differential equation
dy
dx = 3 + 2y
Problem 7. Determine the particular solution
of (y 2 − 1) dy = 3y given that y = 1 when
dx
x = 2 1 6
Rearranging gives:
( y 2 − 1
dx =
3y
Integrating gives:
∫
dx =
) ( y
dy =
3 − 1 )
dy
3y
∫ ( y
3 − 1 3y
)
dy
i.e. x = y2
6 − 1 ln y + c,
3
which is the general solution.
When y = 1, x = 2 1 6 , thus 2 1 6 = 1 6 − 1 3
ln 1 + c, from
which, c = 2.
Hence the particular solution is:
x = y2
6 − 1 3 ln y + 2
Problem 8. (a) The variation of resistance,
R ohms, of an aluminium conductor with
temperature θ ◦ C is given by dR = αR, where
dθ