Problem 47 - Calculus and its applications

Question

(a) write a differential equation that models the situation, and (b) find the general solution. If an initial condition is given, find the particular solution. Recall that when \(y\) is directly proportional to \(x,\) we have \(y=k x\), and when \(y\) is inversely proportional to \(x,\) we have \(y=k / x,\) where \(k\) is the constant of proportionality. In these exercises, let \(k=1\). The rate of change of \(y\) with respect to \(x\) is directly proportional to the square of \(y\).

Step-by-Step Solution

Answer

Differential equation: \(\frac{dy}{dx} = y^2\). General solution: \(y = -\frac{1}{x+C}\).

Step 1: Understand the problem

The rate of change of a variable with respect to another is described by a derivative. In this problem, the rate of change of \(y\) with respect to \(x\) is directly proportional to the square of \(y\). This means we are looking for a differential equation \(\frac{dy}{dx}\) that represents this relationship.

Step 2: Write the differential equation

Since the rate of change of \(y\) with respect to \(x\) is directly proportional to \(y^2\), we write the differential equation as \(\frac{dy}{dx} = y^2\). Given the constant of proportionality, \(k = 1\), is already set, we directly have the equation \(\frac{dy}{dx} = y^2\).

Step 3: Set up the integration for the general solution

To find the general solution to the differential equation, we need to solve \(\frac{dy}{dx} = y^2\). First, separate the variables: \(\frac{dy}{y^2} = dx\).

Step 4: Integrate both sides

Integrate both sides separately. The integration of \(\frac{1}{y^2}\) with respect to \(y\) is \(-\frac{1}{y}\), and the integration of \(1\) with respect to \(x\) is \(x\). The general solution of the differential equation is \(-\frac{1}{y} = x + C\), where \(C\) is the constant of integration.

Step 5: Solve for y to find the general solution

Rearrange the expression \(-\frac{1}{y} = x + C\) to solve for \(y\). This gives \(y = -\frac{1}{x+C}\). This is the general solution of the differential equation.

Key Concepts

Rate of Change

In mathematics, the "Rate of Change" is a measure of how a quantity changes with respect to another variable. Specifically, in the context of differential equations, it refers to how a function changes as its input changes. Often, this rate is expressed with derivatives. For example, the rate of change of a function \(y\) with respect to \(x\) is written as \( \frac{dy}{dx} \), which describes how \(y\) changes for a small change in \(x\).

Understanding the rate at which things change allows us to model real-world phenomena accurately. In our exercise, the rate of change of \(y\) concerning \(x\) is given as directly proportional to \(y^2\). This means the way \(y\) increases or decreases is closely tied to its square, demonstrating a specific dynamic as variables interact. Recognizing this relationship helps us form the equation that represents this rate, which is crucial for solving further.

Proportionality

Proportionality is a concept that depicts how two quantities change in relation to each other. A statement of proportionality might be direct or inverse. When something is directly proportional, it means as one quantity increases, the other one increases at a constant rate. Conversely, when it's inversely proportional, one quantity increases while the other decreases.

In the differential equation from the exercise, we see a direct proportionality where the rate of change of \(y\) is directly tied to \(y^2\). The constant of proportionality in this scenario is considered \(k=1\). This decision simplifies the expression to \( \frac{dy}{dx} = y^2 \). Here, the rate of change of \(y\) scales uniformly with \(y^2\). Proportionality principles are valuable for forming and solving equations that mirror the real-world patterns of the related quantities.

Separation of Variables

"Separation of Variables" is a technique used to solve particular types of differential equations. The idea is to manipulate the equation so that each type of variable is isolated on one side of the equation, allowing for straightforward integration. This step is crucial when dealing with the differential equation \( \frac{dy}{dx} = y^2 \).

The separation process for this equation begins by rewriting it as \( \frac{dy}{y^2} = dx \). Now, each side contains only one type of variable: \(y\) terms on the left, and \(x\) on the right. By integrating both sides independently, it becomes possible to derive a solution that combines both original variables into one cohesive function. This manipulation is a foundational technique for solving simpler differential equations and gaining insight into more complex scenarios.

General Solution

The "General Solution" of a differential equation is a family of functions that encompasses all possible solutions. It is usually accompanied by a constant of integration, which can take any value, thereby offering an infinite number of solutions.

For our exercise, after separating the variables and integrating, we reached the general solution \(-\frac{1}{y} = x + C\), where \(C\) represents the constant of integration. Solving for \(y\) gives \(y = -\frac{1}{x+C}\). This expression includes a range of possible curves, each differing by the value of \(C\).

• Without a specific initial condition, each unique solution remains part of this general set.
• If an initial condition is available (like a specific value of \(y\) when \(x\) is known), it allows us to find a "particular solution," adjusting \(C\) to fit that condition.

This ability to handle various scenarios makes the general solution a powerful tool in differential equations.