IBHM 446 472 .pdf
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16 Integration 3 – Applications
a Following the method of separating variables:
1 dy
1
y dx
x qx
1
冮 y dx dx 冮 x qx dx
1 dy
1
1
冮 y dy 冮 x qx dx
1
冮 y dy 冮 x 11 q2 dx
1
1
1
1
To simplify equations of this
type (i.e. where natural
logarithms appear in all
terms) it is often useful to let
c ln k.
˛
1 ln冨y冨
1
ln冨x冨 c
1q
1 ln冨y冨
1
ln冨x冨 ln k
1q
Now by using the laws of logarithms:
ln冨y冨 ln k
1
ln冨x冨
1q
Technically the absolute value
signs should remain until the
end, but in this situation they
are usually ignored.
y
1
1 ln2 2 ln冨x冨1 q
k
y
1
1 x1 q
k
˛
˛
1
1 y kx1 q
˛
b The curve passes through the point (1, 1) and q 2.
1
1 k 112 1 2
˛
1 1 k 112 1
˛
1k1
1 y 1x1
˛
1y
1
x
Another real-world application of differential equations comes from work done with
kinematics.
Example
A body has an acceleration a, which is dependent on time t and velocity v and is
linked by the equation
a v sin kt
Given that when t 0 seconds, v 1 ms1 and when t 1 second,
v 2 ms1, and that k takes the smallest possible positive value, find the velocity
of the body after 6 seconds.
From the work on kinematics, we know that acceleration is the rate of change
dv
of velocity with respect to time, i.e. a .
dt
Therefore the equation can be rewritten as
454
dv
v sin kt.
dt