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.
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
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 ?
sure, in fact I did it with 235 = 240 e^2k
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?
You have dV/dt = k V
and you know that k = -.010526