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


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