Write a differential equation for the cash balance in the form dB/dt = f[D(t), W(t), i] where t is time (months), D(t) is the deposits as a function of time ($/month), W(t) is the... Write a differential equation for the cash balance in the form dB/dt = f[D(t), W(t), i] where t is time (months), D(t) is the deposits as a function of time ($/month), W(t) is the withdrawals as a function of time ($/month). Assume that interest is compounded continuously; that is, interest iB.
Understand the Problem
The question is asking to derive a differential equation that models the cash balance over time, taking into account the functions of deposits and withdrawals, as well as the interest rate. We need to express this relationship in the specified format.
Answer
The differential equation is $$ \frac{dC(t)}{dt} = D(t) - W(t) + rC(t) $$.
Answer for screen readers
The differential equation that models the cash balance over time is given by:
$$ \frac{dC(t)}{dt} = D(t) - W(t) + rC(t) $$
Steps to Solve
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Define Variables Let ( C(t) ) represent the cash balance at time ( t ). We also need to define ( D(t) ) as the deposit function and ( W(t) ) as the withdrawal function. The interest rate is denoted as ( r ).
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Establish the Change in Cash Balance The change in cash balance over time can be expressed as:
$$ \frac{dC(t)}{dt} = D(t) - W(t) + rC(t) $$
This equation accounts for deposits, withdrawals, and interest earned on the current balance.
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Include Interest Rate Effect Here, ( rC(t) ) represents the income generated from the balance due to the interest rate. Thus the complete formula takes into account not just the inflow and outflow of cash, but also how the cash balance changes due to interest.
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Combine All Elements Combining these concepts, our final differential equation modeling the cash balance over time is:
$$ \frac{dC(t)}{dt} = D(t) - W(t) + rC(t) $$
This describes how the cash balance ( C(t) ) evolves over time.
The differential equation that models the cash balance over time is given by:
$$ \frac{dC(t)}{dt} = D(t) - W(t) + rC(t) $$
More Information
This equation provides a framework for understanding the dynamics of a cash balance in financial contexts, allowing for the analysis of how various deposits, withdrawals, and interest rates impact the overall cash flow.
Tips
- Forgetting to include the interest term ( rC(t) ), which is crucial in modeling the cash balance accurately.
- Confusing deposits and withdrawals in the equation, which can lead to an incorrect sign in the differential equation.
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