Answer :
Answer:
y' = 0.06y -1200; y(0) = 8000
Step-by-step explanation:
You want the differential equation for the investment account balance when it is initially $8000, earns 6% interest continuously, and has a continuous withdrawal at the rate of $1200 per year.
Interest
The interest earned (per year) is 6% of the current balance. If the current balance is represented by y, the interest earned is 0.06y.
Withdrawal
The withdrawal rate is $1200 per year. The corresponding annual rate of change of the account balance is -1200.
Differential equation
The rate of change of the account balance is the sum of the rates of change due to earnings and withdrawals:
y' = 0.06y -1200
In order to solve this differential equation, we must also have an initial condition. That is provided by the initial balance:
y(0) = 8000
The time period used here is years, so the units of y' are dollars per year.
__
Additional comment
As you already know, the solution is ...
y(t) = 20000 -12000e^(0.06t)
To create a differential equation that accounts for continuous deposit and withdrawal, the equation dA/dt = r * A - W is used, where A(t) is the account balance at time t, r is the continuous interest rate (0.06 for 6% annual rate), and W is the withdrawal rate ($1200 per year).
To write a differential equation describing the continuous deposit and withdrawal in an account, we need to incorporate both the continuous interest and the withdrawals. Let's denote the amount of money in the account at time t as A(t).
The interest is paid at a continuous rate of 6% per annum, which is represented mathematically by the constant r = 0.06. The withdrawals are made at a continuous rate of $1200 per year.
The equation for the continuous growth of the investment due to interest is dA/dt = r * A, where dA/dt represents the derivative of A with respect to t, indicating the change in account balance with time, and r is the continuous interest rate.
However, since we're also withdrawing money continuously, we must subtract the withdrawal rate from this expression. Therefore, the differential equation becomes: dA/dt = r * A - W, where W = 1200 is the withdrawal rate.
Putting in the values we have: dA/dt = 0.06 * A - 1200. This equation will describe the behavior of the investment account over time, considering both the continuous compounding interest and the continuous withdrawals.
