(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

a. Assume that y_{o}dollars are deposited into an account paying r percent compounded continuously. If withdrawals are at an annual rate of 200t dollars (assume these are continuous), find the amount in the account after T years.

b. Consider the special case if r = 10% and y_{0}=$20000

c. When will the account be depleted if y_{0}=$5000? Give your answer to the nearest month.

2. Relevant equations

3. The attempt at a solution

I've realized that the rate at which the account balance varies is the following:

dy/dt = ry - 200 (where r is the r percent rate, 0.10; and y the amount of money present)

However, when i try to obtain the differential equation, I keep getting that the amount of money present is the following:

y(T) = 200/r + (y_{0}-200/r)e^{rT}

This would, mean that the function would never decrease in the case of $20000 and as well for $5000 (meaning it will never be depleted). However, I'm pretty sure that i'm wrong on this one. Could anyone please help me with this? My procedure:

1/(ry-200) dy = 1 dt (integrate both parts)

ln(ry-200) 1/r = t + M_{1}

ln(ry-200) = rt + M_{2}

M_{3}e^{rt}=ry-200

y = M_{4}e^{rt}+ 200/r

Then, if y(0) = y_{0}:

y_{0}- 200/r = M_{4}

We then plug this result into our equation:

y = 200/r + (y_{0}- 200/r)e^{rT}

This corresponds to the equation i've been getting. Is my procedure done right?

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# FO Differential equations and account balance

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