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).] You make an initial deposit of $\$ 500$ in a savings account and plan on making future deposits at a gradually increasing annual rate given by $90 t+810$ dollars per year, $t$ years after the initial deposit. Assume that the deposits are made continuously and that interest is compounded continuously at the rate of $6 \% .$ Let $P(t)$ denote the amount of money in the account. (a) Set up an initial-value problem that is satisfied by $P(t).$ (b) Find $P(t).$

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

You make an initial deposit of $\$ 500$ in a savings account and plan on making future deposits at a gradually increasing annual rate given by $90 t+810$ dollars per year, $t$ years after the initial deposit. Assume that the deposits are made continuously and that interest is compounded continuously at the rate of $6 \% .$ Let $P(t)$ denote the amount of money in the account.
(a) Set up an initial-value problem that is satisfied by $P(t).$
(b) Find $P(t).$

Key Concepts

Initial Value Problems

In the context of differential equations, an initial value problem specifies a differential equation together with an initial condition that the solution must satisfy. This approach is fundamental in modeling dynamic systems where the state at a starting time is known, and the differential equation describes how the state evolves over time.

Continuous Compounding and Continuous Deposits

Continuous compounding refers to the mathematical idealization where interest is calculated and added to the principal an infinite number of times per period, leading to exponential growth. When combined with continuous deposits, the model requires integrating a rate function for the deposits over time, reflecting an ongoing inflow of funds into the account.

Solving Linear Differential Equations with Integrating Factors

A first-order linear differential equation can often be solved using an integrating factor, a technique that involves multiplying the differential equation by a specially chosen function to simplify it into an exact derivative. This method is particularly useful for equations where the rate of change of a quantity depends linearly on the quantity itself and an additional non-homogeneous term.

Integration by Parts

Integration by parts is a technique used to integrate the product of two functions. It is especially useful in solving integrals that emerge when integrating the non-homogeneous part of a linear differential equation, such as in financial models where the deposit rate is given by a polynomial function in time.

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