- #1
Geofram
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Homework Statement
S is the balance of a savings account
W is the amount withdrawn per year.
k is a rate percentage of continuous interest per year
1. Solve the differential Equation above.
2. Draw a phase portrait and assess the solution's stability.
3. Assume you have $1,000,000 for retirement. How long would this savings last if the balance grows at 2% per year and you have to withdraw $50,000 a year to live?
Homework Equations
dS/dt = kS - W
The Attempt at a Solution
To solve this I've isolated like terms:
dS/(kS - W) = dt
ln(kS - W)/k = t + C1
ln(kS - W) = t + C2
kS - W = ekt + eC2
kS = C2ekt + W
S = (C2ekt + W) / k
S = C3ekt + W/k
I'm just unsure about what to plug into find out what happens to the solution for the phase diagram, and also I'm lost on how to solve #3.
For number 3, I would plug in 0 for S because we want to know when it runs out.
k = .02, t = ?, and W = 50,000
I'm just lost on how I get C3. I can plug in the starting values to get:
t = 0 because we're just starting?
1,000,000 = C3e0.02(0) + 50,000 / .02
-1,500,000 = C3e0
C3 = -1,500,000
But if I plug this back in:
0 = (-1,500,000)e0.02t + 50,000/.02
(-2,500,000)/(-1,500,000) = e0.02t
ln(1.6667) = 0.02t
t = 25.54 years?
That actually makes sense...
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