Solving the differential equations that arise from modeling...

Solving the differential equations that arise from modeling may require using integration by parts. [See formula (1).] A person deposits an inheritance of $\$ 100,000$ in a savings account that earns $4 \%$ interest compounded continuously. This person intends to make withdrawals that will increase gradually in size with time. Suppose that the annual rate of withdrawals is $2000+500 t$ dollars per year, $t$ years from the time the account was opened. (a) Assume that the withdrawals are made at a continuous rate. Set up a differential equation that is satisfied by the amount $f(t)$ in the account at time $t.$ (b) Determine $f(t).$ (c) With the help of your calculator, plot $f(t)$ and approximate the time it will take before the account is depleted.

Solving the differential equations that arise from modeling may require using integration by parts. [See formula (1).]

A person deposits an inheritance of $\$ 100,000$ in a savings account that earns $4 \%$ interest compounded continuously. This person intends to make withdrawals that will increase gradually in size with time. Suppose that the annual rate of withdrawals is $2000+500 t$ dollars per year, $t$ years from the time the account was opened.
(a) Assume that the withdrawals are made at a continuous rate. Set up a differential equation that is satisfied by the amount $f(t)$ in the account at time $t.$
(b) Determine $f(t).$
(c) With the help of your calculator, plot $f(t)$ and approximate the time it will take before the account is depleted.

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Key Concepts

Integration by Parts

Integration by parts is a calculus technique derived from the product rule of differentiation. It is used to integrate products of functions and is particularly useful when solving the integrals that arise during the process of solving differential equations in more complex systems.

Differential Equations in Financial Modeling

Differential equations are used to model dynamic systems and quantify changes over time. In financial contexts, they capture the interplay of factors like interest accrual and withdrawals, allowing us to predict the evolution of account balances or investments under continuous changes.

Continuous Compounding Interest

Continuous compounding refers to the process of earning interest on an investment where the accumulation occurs at every possible moment. This concept leads to exponential growth models and is often expressed mathematically as a differential equation, providing a more precise depiction of interest accumulation over time.

Integration Factor Technique

The integration factor technique is a method for solving first-order linear differential equations. By multiplying the differential equation by a strategically chosen function, the equation becomes exact or easier to integrate, ultimately leading to a solution that incorporates initial conditions and system parameters.

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