Problem 56 - Calculus for Biology and Medicine

Question

Suppose that \(x(t)+y(t)\) is a conserved quantity. If $$ \frac{d x}{d t}=-3 x+2 x y $$ find the differential equation for \(y(t)\).

Step-by-Step Solution

Answer

The differential equation for \(y(t)\) is \(\frac{d y}{d t} = 3x - 2xy\).

Step 1: Understand the Given Information

We have been given that the quantity \(x(t) + y(t)\) is conserved, which means it is a constant over time. This implies that its derivative with respect to time \(\frac{d}{dt}(x+y) = 0\).

Step 2: Use the Conservation Law

Since \(x + y\) is conserved, we have:\[ \frac{d}{dt}(x + y) = \frac{d x}{d t} + \frac{d y}{d t} = 0. \]This means:\[ \frac{d x}{d t} = -\frac{d y}{d t}. \]

Step 3: Substitute the Given Derivative

We know \(\frac{d x}{d t} = -3x + 2xy\), and from the conservation law, \(\frac{d y}{d t} = -\frac{d x}{d t}\). Substitute \(\frac{d x}{d t}\) into this equation:\[ \frac{d y}{d t} = 3x - 2xy. \]

Step 4: Write the Differential Equation for \(y(t)\)

We have derived the differential equation for \(y(t)\):\[ \frac{d y}{d t} = 3x - 2xy. \]This is the required differential equation for \(y(t)\).

Key Concepts

Understanding Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In our context, we often deal with time-dependent processes, so we represent these relationships as:

• The function, like our variables \(x(t)\) and \(y(t)\), which can change over time.
• The derivatives, which express how fast and in what direction these functions change over time.

Differential equations are essential because they model real-world phenomena, ranging from physics to biology.
In this exercise, we encounter a differential equation that describes how the variable \(x(t)\) changes, given by \(\frac{d x}{d t}=-3 x+2 x y\). This equation helps us understand the dynamics of system where \(x(t) + y(t)\) is conserved.
To solve these problems, we usually need to find a function that satisfies the equation, often requiring integration and more advanced techniques, depending on the complexity of the system involved.

Exploring Conserved Quantities

Conserved quantities are fundamental concepts in science. A conserved quantity remains constant through time, even as the system evolves. In our exercise, we have \(x(t) + y(t)\) as a conserved quantity, meaning:

• \(\frac{d}{dt}(x+y) = 0\): The rate of change of \(x(t) + y(t)\) is zero, confirming its constant nature over time.
• This stems from physical laws like conservation of energy or mass, where parts may change, but the total remains constant.

Our task involves utilizing these principles to control how \(y(t)\) behaves as \(x(t)\) changes.
By understanding that the conservation law states \( \frac{d x}{d t} + \frac{d y}{d t} = 0 \), we derive how changes in \(x(t)\) directly influence changes in \(y(t)\), maintaining the conserved sum. Finding the correct relationship or differential equation for \(y(t)\) requires careful substitution and manipulation of given information.

The Role of Derivatives

Derivatives are essential tools in calculus, capturing how a function changes at any point. In differential equations, derivatives allow us to explore the rate of change of variables over time. Specifically:

• They serve as the backbone in formulating relationships between quantities that evolve.
• In our scenario, \(\frac{d x}{d t} = -3x + 2xy\) specifies the rate at which \(x(t)\) changes.
• Similarly, \(\frac{d y}{d t}\) derived from conservation principles identifies how \(y(t)\) changes.

Using derivatives, we can build and solve equations leading to explicit solutions or insights about how the system behaves.
They help us connect changes in one variable to another, as demonstrated when we derived \(\frac{d y}{d t} = 3x - 2xy\). This equation reflects how \(y(t)\) must adjust to ensure the total \(x(t) + y(t)\) remains conserved across time.