Trying to apply my DiffEq solution.

[Geofram]

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W is the amount withdrawn per year.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 = e^kt + e^C2

kS = C2e^kt + W

S = (C2e^kt + W) / k

S = C3e^kt + 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 = C3e^0.02(0) + 50,000 / .02
-1,500,000 = C3e⁰
C3 = -1,500,000

But if I plug this back in:

0 = (-1,500,000)e^0.02t + 50,000/.02
(-2,500,000)/(-1,500,000) = e^0.02t
ln(1.6667) = 0.02t
t = 25.54 years?

That actually makes sense...

 

[HallsofIvy]

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W is the amount withdrawn per year.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 = e^kt + e^C2

You went wrong here.
$$e^{t+ C_2}= (e^t)(e^{C_2}$$
the product of the exponentials, not the sum. Also, there should be no "k" multiplying t.

kS = C2e^kt + W

S = (C2e^kt + W) / k

S = C3e^kt + 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.

 

[Geofram]

ln(kS - W)/k = t + C1

ln(kS - W) = t + C2

Actually I just forgot to do it in the step I multiplied k out.
It should be:

ln(kS - W)/k = t + C1

ln(kS - W) = kt + C2

Take a look at what I added, does that seem right? I'm still confused on how to accomplish #2