# Deriving a differential equation for a loan/interest problem

Tags:
1. Sep 29, 2011

### JNBirDy

1. The problem statement, all variables and given/known data
You borrow money from a friend at a continuous interest rate of r% per month. You want to pay your friend back as quickly as you can at the beginning, but reduce your payment rate over time. You decide to pay off at a continuously decreasing rate given by K₀e$^{-at}$, in dollars per month.

Write a differential equation that describes how much you owe and solve it.

2. Relevant equations
None

3. The attempt at a solution
Let S be the amount borrowed -

dS/dt = rS - K₀e$^{-at}$

S' - rS = -K₀e$^{-at}$

S'(I(x)) - rS(I(x)) = -K₀e$^{-at}$(I(x))

Se$^{-rt}$ = -K₀$\int$e$^{-t(a+r)}$

Se$^{-rt}$ = ...

This is where I get stuck, I have don't understand how to integrate -K₀$\int$e$^{-t(a+r)}$, any hints?

2. Sep 29, 2011

### rock.freak667

if 'a' and 'r' are constants then you can simply recall that ∫ ekt = (1/k)ekt+ constant.