Suppose $B(t)$ is the amount of money in the account (or owed to the bank) at time $t$. Then $B(0)=B_0$ is the initial balance,
and the rate of change of the balance is
$$
\frac{dB}{dt}=r B(t)+m
$$
Here
$r$ is the interest rate under continuous compounding(nobody compounds continuously!)
$m$ is the payment (or withdrawal, if negative) amount per unit of time under continuous payment(nobody pays continuously!)
Warning about units: 1 year = 12 months Example: If the annual interest rate is 12% and someone saves 100USD per month, then the balance solves the linear and separable equation which depend on the choice of units of time.
$B'=.12 B+1200$, if time $t$ is measured in years
$B'=.01B+100$ if time $t$ is measured in month
$
\frac{dB}{dt}=r B(t)+m, \; B(0)=B_0$
DE Poll
Work out your answer on paper or using appropriate software. Then put your answer into Webex Poll.
https://www.wolframalpha.com/examples/mathematics/differential-equations/
https://www.symbolab.com/solver/ordinary-differential-equation-calculator
A CCCH has a 5,000 balance on a credit card that carries 12% annual interest rate. The CCCH is making the minimal monthly payments of 25.
Determine after how many years the CCCH will have a balance of 1,000,000=$10^6$ on this credit card. Put your answer into DE-Poll using two decimals
The answer is $\approx 50$.
Solution Yearly payments are $25\times 12=300$ so we want to solve differential equation
$B'=.12 B- 300$ with the initial condition $B(0)=5000$. WolframAlpha
gives the formula $B(t)=2500 + 2500 e^{0.12 t}$. We want to solve equation $B(t)=10^6$ for $t$.
WolframAlpha says $t=25/3 (\log(3) + \log(7) + \log(19))\approx 49.91$