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 owed to the bank at the end of the each day after the loan closing.


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

$Mr and Mr young borrowed Z amount of money from$ $a bank at a daily interest rate r . The youngs make the$ $same micro payment of D amount to the bank each day .$ $Set up a differential equation for amount x owed$ $to the bank at the end of the each day after the loan$ $closing .$
Commented by MWSuSon last updated on 08/Feb/22

$How do i set up a differential equation concerning$ $this question sir ?$
Commented by mr W last updated on 08/Feb/22

$t h e s a m e a s Q 165404$
Commented by mr W last updated on 08/Feb/22

$l o a n : Z$ $d a i l y i n t e r e s t r a t e : r$ $d a i l y p a y b a c k : D$ $a f t e r t d a y s t h e Y o u n g s o w e t h e b a n k$ $m o n e y x .$ $t a s d i s c r e t e v a r i a b l e :$ $f r o m Q 165404 w e h a v e$ $x = Z {(1 + r)}^{t} - \frac{D [{(1 + r)}^{t} - 1]}{r}$ $x = \frac{D}{r} + (Z - \frac{D}{r}) {(1 + r)}^{t}$
Commented by mr W last updated on 08/Feb/22

$l o a n : Z$ $d a i l y i n t e r e s t r a t e : r$ $d a i l y p a y b a c k : D$ $a f t e r t d a y s t h e Y o u n g s o w e t h e b a n k$ $m o n e y x .$ $t a s c o n t i n o u s v a r i a b l e :$ $d x = (r x - D) d t$ $\frac{d x}{d t} = r x - D \leftarrow d i f f e r e n t i a l e q n .$ $\frac{d x}{r x - D} = d t$ $\ln (r x - D) = r t + C_{1}$ $\Rightarrow x = \frac{D}{r} + {C e}^{r t}$ $a t t = 0 :$ $\Rightarrow Z = \frac{D}{r} + C$ $\Rightarrow x = \frac{D}{r} + (Z - \frac{D}{r}) e^{r t}$
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

$t h a n k s s i r!$ $t h i s e x a m p l e s h o w s e x a c t l y h o w t h e$ $n u m b e r ‘ ‘ e^{″} w a s d i s c o v e r e d . w h e n$ $w e c a l c u l a t e t h e i n t e r e s t n o t d a i l y,$ $e . g . w e d i v i d e a d a y i n t o m a n y p a r t s,$ $t h e n w e g e t$ $x = \frac{D}{r} + (Z - \frac{D}{r}) {(1 + \frac{r}{n})}^{n t}$ $x = \frac{D}{r} + (Z - \frac{D}{r}) {[{(1 + \frac{r}{n})}^{\frac{n}{r}}]}^{r t}$ $w h e n w e d i v i d e a d a y i n t o i n f i n i t e$ $m a n y p a r t s, i . e . n \to \infty, t h e n$ $x = \frac{D}{r} + (Z - \frac{D}{r}) {[\lim_{n \to \infty} {(1 + \frac{r}{n})}^{\frac{n}{r}}]}^{r t}$ $p e o p l e f o u n d t h a t t h i s n u m b e r$ $\lim_{n \to \infty} {(1 + \frac{r}{n})}^{\frac{n}{r}} r e a l l y e x i s t s, a n d$ $d e n o t e d i t a s ‘ ‘ e^{″} . a n d w e g e t$ $x = \frac{D}{r} + (Z - \frac{D}{r}) e^{r t}$
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

$Weldone sir$
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