Solution of first order differential equations by separation of variables 447
5.
1
e x + 2 = x − 3
d y
dx
, given y = 1 when x = 0.
y =
1
6
x 2 − 4x +
2
e x + 4
6. The gradient of a curve is given by:
d y
dx
+
x 2
2
= 3x
Find the equation of the curve if it passes
through the point
1,
1
3
.
y =
3
2
x 2 −
x 3
6
− 1
7. The acceleration, a, of a body is equal to its rate
of change of velocity,
dv
dt
. Find an equation for
v in terms of t , given that when t = 0, 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
dt
= u − gt , where s is the height of the object
in metres at time t seconds and g = 9.8 m/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
d y
dx
= f ( y) is initially
rearranged to give dx =
d y
f ( y)
and then the solution is
obtained by direct integration,
i.e.
dx =
dy
f ( y)
Problem 6. Find the general solution of
d y
dx
= 3 + 2y.
Rearranging
d y
dx
= 3 + 2y gives:
dx =
d y
3 + 2y
Integrating both sides gives:
dx =
d y
3 + 2y
Thus, by using the substitution u = (3 + 2y) — see
Chapter 39,
x =
1
2 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 =
1
2 ln(3 + 2y) + ln k,
i.e. x = ln(3 + 2y)
1
2 + 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
d y
dx
= 3 + 2y
Problem 7. Determine the particular solution of
( y
2
− 1)
d y
dx
= 3y given that y = 1 when x = 2
1
6
Rearranging gives:
dx =
y 2 − 1
3y
d y =
y
3
−
1
3y
d y
Integrating gives:
dx =
y
3
−
1
3y
d y
i.e.
x =
y
2
6
−
1
3
ln y + c,
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 =
y 2
6
−
1
3
ln y + 2
5.
1
e x + 2 = x − 3
d y
dx
, given y = 1 when x = 0.
y =
1
6
x 2 − 4x +
2
e x + 4
6. The gradient of a curve is given by:
d y
dx
+
x 2
2
= 3x
Find the equation of the curve if it passes
through the point
1,
1
3
.
y =
3
2
x 2 −
x 3
6
− 1
7. The acceleration, a, of a body is equal to its rate
of change of velocity,
dv
dt
. Find an equation for
v in terms of t , given that when t = 0, 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
dt
= u − gt , where s is the height of the object
in metres at time t seconds and g = 9.8 m/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
d y
dx
= f ( y) is initially
rearranged to give dx =
d y
f ( y)
and then the solution is
obtained by direct integration,
i.e.
dx =
dy
f ( y)
Problem 6. Find the general solution of
d y
dx
= 3 + 2y.
Rearranging
d y
dx
= 3 + 2y gives:
dx =
d y
3 + 2y
Integrating both sides gives:
dx =
d y
3 + 2y
Thus, by using the substitution u = (3 + 2y) — see
Chapter 39,
x =
1
2 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 =
1
2 ln(3 + 2y) + ln k,
i.e. x = ln(3 + 2y)
1
2 + 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
d y
dx
= 3 + 2y
Problem 7. Determine the particular solution of
( y
2
− 1)
d y
dx
= 3y given that y = 1 when x = 2
1
6
Rearranging gives:
dx =
y 2 − 1
3y
d y =
y
3
−
1
3y
d y
Integrating gives:
dx =
y
3
−
1
3y
d y
i.e.
x =
y
2
6
−
1
3
ln y + c,
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 =
y 2
6
−
1
3
ln y + 2