In general, one uses differential equations (and the methods we have developed for their solution) when a function is described by conditions on its rate of change, but one wishes to find a closed form expression for the function.
An object falling in a vacuum subject to a constant gravitational force accelerates at a constant rate.
If the object were to be dropped from rest and to attain
a velocity of after one second, how fast would it be
traveling after five seconds?
Let be the velocity at time
seconds measured in meters per second.
Then we know that
, that
, and that
(the acceleration,
the rate of change of the velocity, so
, is constant).
Integrating the equation with respect to
, we see that
.
Thus, if
, we have
. Integrating again, we see that
. Setting
, we have
.
Evaluating at and
we have
and
. Thus,
so that
.
An object performing a free fall subject to a constant gravitational force in a viscous fluid is slowed by a drag which is proportional to its velocity.
Find a general expression for the velocity of such an object.
Let be the constant rate of gravitational acceleration,
the constant of proportionality for the drag force, and
the
initial velocity.
Then the velocity, , satisfies
In symbols,
or
.
This is a linear first order differential equation which we may solve using the method integration factors.
Here and
.
So,
.
A loan has a fixed interest rate of % (the interest is
compounded continuously) and the borrower repays the loan at a
constant rate of $10,000 dollars per year. If the initial value
of the loan was $100,000, when will the debt be retired?
Let be the remaining principal at time
years. We were told
that
and we wish to find
so that
.
As
, we see that
.
So, when
or
.