Problem 9 - 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? $$ \frac{d y}{d t}=-4 y, \quad y=30 \text { when } t=0 $$

Step-by-Step Solution

Answer

The exact solution is \(y(t) = 30e^{-4t}\), which satisfies the equation \(\frac{d y}{d t} = -4 y\). The function represents exponential decay.

Step 1: Write the Differential Equation

From the given information, it's clear that the differential equation is \(\frac{d y}{d t}= -4y\)

Step 2: Solve the Differential Equation

The general solution of the differential equation \(\frac{d y}{d t}= -4y\) is given by \(y(t) = A e^{-4t}\), where A is an arbitrary constant which can be determined by initial conditions.

Step 3: Use Initial Condition to Find Value of A

Using the initial condition given in the question that \(y(0) = 30\), we can substitute it into the general solution of our differential equation to find the value of the constant A. So, substituting: \(30 = A * e^{-4*0}\), we find that \(A = 30\). Hence, the exact solution is: \(y(t) = 30e^{-4t}\).

Step 4: Confirm Analytically

Deriving the right hand side of the function \(y(t) = 30e^{-4t}\) we get \(\frac{d y}{d t} = -120e^{-4t}\), which is equal to \( -4 * y(t)\). Therefore the function satisfies the equation \(\frac{d y}{d t} = -4 y\).

Step 5: Identify Exponential Growth or Decay

Since the coefficient of the function is negative, this function represents exponential decay.

Key Concepts

Exponential Growth and Decay

In the realm of differential equations, understanding whether a function embodies exponential growth or decay is crucial. An equation of the form \( \frac{d y}{d t} = C y \) represents exponential behavior, where the constant \( C \) determines the nature of this behavior. If \( C \) is positive, the function represents exponential growth—meaning the function's value will increase exponentially over time. However, if \( C \) is negative, as it is in our problem \( \frac{d y}{d t} = -4y \), the function demonstrates exponential decay. This implies that the value of the function will decrease exponentially over time.
Exponential decay is characterized by a rapid decrease at first, which slows down as time progresses. In our solution, you observe this through the exponential term \( e^{-4t} \), which makes the product decrease as \( t \) increases.
In practical terms, this could model scenarios such as radioactive decay or the cooling of a hot object, where the rate of change of a quantity is proportional to its current value, leading to a steady decrease over time.

Initial Conditions

Initial conditions in differential equations are the specific values that allow us to find particular solutions. They are essential for determining the unknown constants that arise when solving these equations. In the exercise, the initial condition is given as \( y = 30 \) when \( t = 0 \). This information is used to compute the particular solution for the differential equation.
Simply put, initial conditions help us tailor the generic or general solution of a differential equation to fit a specific situation or scenario. In our problem, when we solved the equation \( y(t) = A e^{-4t} \), the constant \( A \) was yet to be determined until we used the initial condition.
By substituting the initial values into the general solution \( 30 = A e^{0} \), we discovered that \( A = 30 \). This enabled us to write the exact solution as \( y(t) = 30 e^{-4t} \), uniquely fitting the situation described by the initial condition.

Analytical Solution

An analytical solution involves finding a closed-form expression that describes the function entirely, rather than a numerical or approximated solution. In our exercise, we derived an analytical solution to the differential equation \( \frac{d y}{d t}= -4y \) by methodically working through the problem.
To confirm the analytical solution, we verified that the derived expression \( y(t) = 30 e^{-4t} \) satisfies the original differential equation. This is essential to ensure the correctness and reliability of our solution. We calculated the derivative \( \frac{d y}{d t} \) and confirmed it equaled the right side of the equation, proving that our solution was indeed correct.
Analytical solutions offer significant insight into the behavior of the function over time, which can be incredibly useful without relying on computational methods. In summary, always checking that the derived solution satisfies the given differential equation is a vital step in solving such problems analytically.