Savings Account Let A(t) be the balance in a savings account...

Savings Account Let $A(t)$ be the balance in a savings account after $t$ years, and suppose that $A(t)$ satisfies the differential equation $$ A^{\prime}(t)=.045 A(t), \quad A(0)=3000 $$ (a) How much money was originally deposited in the account? (b) What interest rate is being earned? (c) Find the formula for $A(t)$ (d) What is the balance after 5 years? (e) Use part (d) and the differential equation to determine how fast the balance is growing after 5 years. (f) How large will the balance be when it is growing at the rate of $\$ 270$ per year?

Savings Account Let $A(t)$ be the balance in a savings account after $t$ years, and suppose that $A(t)$ satisfies the differential equation
$$
A^{\prime}(t)=.045 A(t), \quad A(0)=3000
$$
(a) How much money was originally deposited in the account?
(b) What interest rate is being earned?
(c) Find the formula for $A(t)$
(d) What is the balance after 5 years?
(e) Use part (d) and the differential equation to determine how fast the balance is growing after 5 years.
(f) How large will the balance be when it is growing at the rate of $\$ 270$ per year?

Key Concepts

Initial Value Problem

An initial value problem involves a differential equation together with a specified value, called the initial condition, at which the solution is known. This allows for a unique solution to be determined for the differential equation by ensuring that the constants of integration are specifically set to satisfy the initial condition.

Exponential Growth Model

An exponential growth model describes processes where the rate of change of a quantity is proportional to the quantity itself. This leads to solutions in the form of exponentials, which are commonly used in applications such as population growth, finance with continuously compounded interest, and radioactive decay.

First-Order Linear Differential Equations

A first-order linear differential equation involves a function and its first derivative and is typically expressed in the form y' = ky, where k is a constant. Such equations are simple yet powerful models in various fields, and their solutions prominently feature exponential functions.

Rate of Change

The concept of rate of change describes how a quantity varies over time. In the context of differential equations, the derivative represents this rate, indicating how quickly the underlying function is increasing or decreasing. This is key in understanding the dynamics of systems modeled by such equations.

Interpreting Differential Equations in Financial Contexts

In financial mathematics, differential equations are used to model the growth of investments, where the balance changes over time according to a rate that is proportional to the current balance. This interpretation helps in calculating accumulated amounts, interest earned, and determining the growth behavior at specific points in time.

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