Question

The median value of a home in a particular market is decreasing exponentially. If the value of a home
was initially $240,000, then its value two years later is $235,000.

a) Write a differential equation that models this situation. Let V represent the value of the home (in thousands of dollars) and t represent the number of years since its value was $240,000.

For this, I got dV/dt = kV but I'm not positive it's correct. I also need to solve for the particular solution in terms of V and t and I'm not sure how to do that.

Answer 1

dV/dt = k V

dV/V = k dt

ln V = k t + C

e^ln V = V definition of ln

V = e^(kt + C)
C is arbitrary so far
V = e^kt * e^C
= C e^kt since e^c could be any old C
Now that is general solution. Now put in t = 0
at t = 0 e^kt = e^0 = 1
so
240,000 = C e^0 = C
so
V = 240,000 e^kt
Now if t = 2, V = 235,000
235 = 240 e^2k

e^2k = .979
ln e^2k = 2k = -.02105
so
k = - .010526
and
V = 240,000 e^-.010526 t

Answer 2

If the question says to write the differential equation "in thousands of dollars" would the particular solution be written as V(t)=240e^((-.010526)(t)) instead of V = 240,000 e^-.010526 t ?

Answer 3

sure, in fact I did it with 235 = 240 e^2k

Answer 4

The last question for this problem asks to find the relative rate of change in the home's value at any time t. How would I do that?

Answer 5

You have dV/dt = k V
and you know that k = -.010526

Answer 6

So would it be -.010526 = (dV/dt)/V ?

Answer 7

I suppose :)