differential equations #2

noscbs · #1 $Old$ November 1st, 2009, 11:08 PM

a certain small country has $10billion in paper currency in circulation, and each day $50million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever the old currency comes into the banks. Let x=x(t) denote the amount of new currency in circulation at time t, with x(0)= 0.

a) formulate a mathematical model in the form of an initial-value problem that represents the flow of the new currency into circulation.
b) solve the initial-value problem found in part (a)
c) How long will it take for the new bills to account for 90% of the currency in circulation?

chisigma · #2 $Old$ November 4th, 2009, 01:22 AM

We hypotize the 'cash flow' of the Central Bank is constant, so that each day $50 milions come in and $50 milions go out, and $50 milions are the entire circulating capital multiplied by .005. Now we set...

$\alpha= \frac{new corrency circulating capital}{total circulating capital}$ (1)

The DE for $\alpha$ is...

$\frac{d \alpha}{dt} = .005 \cdot (1-\alpha)$ , $\alpha(0)=0$ (2)

... where tha variables are separable. The solution is...

$\frac{d \alpha}{1-\alpha} = .005\cdot dt \rightarrow \ln (1-\alpha) = -.005\cdot t + \ln c \rightarrow$

$\rightarrow \alpha = 1-c\cdot e^{-.005\cdot t}$ (3)

With the 'initial condition' $\alpha(0)=0$ is $c=1$ so that is...

$\alpha = 1- e^{-.005\cdot t}$ (4)

It will be $\alpha=.9$ at the time $t=460$ $days$ ...

Kind regards

$\chi$ $\sigma$