Mr-and-Mr-young-borrowed-Z-amount-of-money-from-a-bank-at-a-daily-interest-rate-r-The-youngs-make-the-same-micro-payment-of-D-amount-to-the-bank-each-day-Set-up-a-differential-equation-for-amount-x

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Question Number 165780 by MWSuSon last updated on 08/Feb/22

$$\mathrm{Mr}\mathrm{and}\mathrm{Mr}\mathrm{young}\mathrm{borrowed}\mathrm{Z}\mathrm{amount}\mathrm{of}\mathrm{money}\mathrm{from}$$$$\mathrm{a}\mathrm{bank}\mathrm{at}\mathrm{a}\mathrm{daily}\mathrm{interest}\mathrm{rate}\mathrm{r}.\mathrm{The}\mathrm{youngs}\mathrm{make}\mathrm{the}$$$$\mathrm{same}\mathrm{micro}\mathrm{payment}\mathrm{of}\mathrm{D}\mathrm{amount}\mathrm{to}\mathrm{the}\mathrm{bank}\mathrm{each}\mathrm{day}.$$$$\mathrm{Set}\mathrm{up}\mathrm{a}\mathrm{differential}\mathrm{equation}\mathrm{for}\mathrm{amount}\mathrm{x}\mathrm{owed}$$$$\mathrm{to}\mathrm{the}\mathrm{bank}\mathrm{at}\mathrm{the}\mathrm{end}\mathrm{of}\mathrm{the}\mathrm{each}\mathrm{day}\mathrm{after}\mathrm{the}\mathrm{loan}$$$$\mathrm{closing}.$$

Commented by MWSuSon last updated on 08/Feb/22

$$\mathrm{How}\mathrm{do}\mathrm{i}\mathrm{set}\mathrm{up}\mathrm{a}\mathrm{differential}\mathrm{equation}\mathrm{concerning}$$$$\mathrm{this}\mathrm{question}\mathrm{sir}?$$

Commented by mr W last updated on 08/Feb/22

$$thesameasQ165404$$

Commented by mr W last updated on 08/Feb/22

$$loan:Z$$$$dailyinterestrate:r$$$$dailypayback:D$$$$aftertdaystheYoungsowethebank$$$$moneyx.$$$$$$$$tasdiscretevariable:$$$$fromQ165404wehave$$$$x=Z{(1+r)}^t-\frac{D[{(1+r)}^t-1]}{r}$$$$x=\frac{D}{r}+(Z-\frac{D}{r}){(1+r)}^t$$

Commented by mr W last updated on 08/Feb/22

$$loan:Z$$$$dailyinterestrate:r$$$$dailypayback:D$$$$aftertdaystheYoungsowethebank$$$$moneyx.$$$$$$$$tascontinousvariable:$$$$dx=(rx-D)dt$$$$\frac{dx}{dt}=rx-D\leftarrow differentialeqn.$$$$\frac{dx}{rx-D}=dt$$$$\ln (rx-D)=rt+C_1$$$$\Rightarrow x=\frac{D}{r}+{Ce}^{rt}$$$$att=0:$$$$\Rightarrow Z=\frac{D}{r}+C$$$$\Rightarrow x=\frac{D}{r}+(Z-\frac{D}{r})e^{rt}$$

Commented by MWSuSon last updated on 08/Feb/22

Thank you sir, you have been of wonderful help. i wish you success in all you do sir, thank you.

Commented by mr W last updated on 09/Feb/22

$$thankssir!$$$$thisexampleshowsexactlyhowthe$$$$number“e^{″}wasdiscovered.when$$$$wecalculatetheinterestnotdaily,$$$$e.g.wedivideadayintomanyparts,$$$$thenweget$$$$x=\frac{D}{r}+(Z-\frac{D}{r}){(1+\frac{r}{n})}^{nt}$$$$x=\frac{D}{r}+(Z-\frac{D}{r}){\left[{(1+\frac{r}{n})}^{\frac{n}{r}}\right]}^{rt}$$$$whenwedivideadayintoinfinite$$$$manyparts,i.e.n\to \mathrm{\infty },then$$$$x=\frac{D}{r}+(Z-\frac{D}{r}){\left[\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{r}{n})}^{\frac{n}{r}}\right]}^{rt}$$$$peoplefoundthatthisnumber$$$$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{r}{n})}^{\frac{n}{r}}reallyexists,and$$$$denoteditas“e^{″}.andweget$$$$x=\frac{D}{r}+(Z-\frac{D}{r})e^{rt}$$

Commented by MWSuSon last updated on 09/Feb/22

I don't know why I don't get notified whenever there's a reply on my post, thank you sir for this additional info.

Commented by Tawa11 last updated on 10/Feb/22

$$\mathrm{Weldone}\mathrm{sir}$$

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