Problem 10 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use the given information to write an equation for \(y\). Confirm your result analytically by showing that the function satisfies the equation \(d y / d t=C y .\) Does the function represent exponential growth or exponential decay? $$ \frac{d y}{d t}=5.2 y, \quad y=18 \text { when } t=0 $$

Step-by-Step Solution

Answer

The equation for \( y \) is \( y(t) = 18e^{5.2t} \), and the function represents exponential growth.

Step 1: Write Down Exponential Function Using Initial Conditions

We will start off by writing the general form of an exponential function, \( y(t) = Ae^{rt} \), where \( A \) is the initial value of \( y \), \( r \) is the growth or decay rate and \( t \) is time. Using the given initial condition \( y = 18 \) when \( t = 0 \), the constant \( A \) can be found. Substituting \( t = 0 \) and \( y = 18 \) in, we get \( 18 = Ae^{r*0} = Ae^0 = A \). Hence, \( A = 18 \) and the exponential function becomes \( y(t) = 18e^{rt} \).

Step 2: Obtain the Value of r

To find the value of \( r \), we can substitute \( y(t) \) into the given differential equation and solve for \( r \). Using the derivative form, \( y'(t) = 18*r*e^{rt} \), we compare this with the derivative equation provided, \( \frac{dy}{dt} = 5.2y = 5.2*18e^{rt} \). It is clear that \( r = 5.2 \). Thus, the equation can be written as \( y(t) = 18e^{5.2t} \).

Step 3: Verifying the Solution

To ensure correctness of solution, substitute \( y(t) = 18e^{5.2t} \) into the given differential equation \( \frac{dy}{dt} = 5.2y \) and check that it holds. Let's calculate \( \frac{dy}{dt} = 5.2*18e^{5.2t} = 5.2*y(t) \), which verifies the correctness.

Step 4: Identify nature of the exponential function

Since the \( r \) value in our function \( y(t) = 18e^{5.2t} \) is positive, this function represents exponential growth. In exponential functions, a positive rate symbolizes growth, while a negative one signifies decay. Hence, our function symbolizes exponential growth.

Key Concepts

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. Such equations play a crucial role in understanding how different quantities change over time. They appear frequently in real-world applications, from physics to economics.
In our problem, we have the differential equation \( \frac{dy}{dt} = 5.2y \). This is a first-order linear differential equation, meaning it involves the first derivative of a function \( y \), and this derivative is proportional to \( y \) itself. The constant of proportionality here is 5.2.

To solve this type of equation, we look for a function that, when differentiated, results in a term proportional to itself. This leads us to exponential functions, natural candidates for such equations.
Understanding differential equations helps in recognizing patterns of change and modeling them accurately. It's a cornerstone to solving initial value problems and understanding dynamic systems.

Exponential Growth

Exponential growth occurs when the growth rate of a quantity is proportional to its current size. The larger the quantity gets, the faster it grows. This type of growth is modeled by exponential functions.
If the rate \( r \) in the equation \( y(t) = Ae^{rt} \) is positive, the function exhibits exponential growth. In our example, \( r = 5.2 \), which is positive. Thus, this confirms that the function \( y(t) = 18e^{5.2t} \) is a model of exponential growth.

• Exponential growth is often observed in scenarios involving population increase, compound interest, and more.
• The characteristic feature is the rapid increase as time progresses.

Exponential growth can lead to substantial increases over time, making it an essential concept in various scientific and financial fields.

Initial Value Problems

Initial value problems are a kind of differential equation where you are given the initial state of the system. This initial condition allows you to find a specific solution to a differential equation.
In this problem, we were given \( y = 18 \) when \( t = 0 \). This is our initial condition, which helps in determining the constant \( A \) in the general solution \( y(t) = Ae^{rt} \).
The process involves:

• Finding the constant \( A \) using the initial condition.
• Determining the nature of the function by finding \( r \).

Once you establish the correct initial condition and solve the differential equation, you form the precise solution that models the system's behavior from that initial point forward. Thus, initial value problems are crucial in scenarios where the starting state significantly influences future development.