Ben buys a car for $50,000. The value of the car decreases at a rate of 4% per year. How much will the car be worth in 3 years?
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From the homogeneous part of the ODE, we can get two fundamental solutions. The characteristic equation is
which has roots at
. This admits the two fundamental solutions
The particular solution is easiest to obtain via variation of parameters. We're looking for a solution of the form
where
and
is the Wronskian of the fundamental solutions. We have
and so
Therefore the particular solution is
so that the general solution to the ODE is
which has roots at
The particular solution is easiest to obtain via variation of parameters. We're looking for a solution of the form
where
and
and so
Therefore the particular solution is
so that the general solution to the ODE is