Problem 21 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. Every differential equation of the form \(y^{\prime}=f(y)\) is separable.

Step-by-Step Solution

Answer

True, every differential equation of the form \( y' = f(y) \) is separable.

Step 1: Understanding the Problem

We need to determine if every differential equation of the form \( y' = f(y) \) is separable. A separable differential equation can be rearranged to \( g(y)dy = h(x)dx \). This implies that all terms involving \( y \) can be placed on one side of the equation and those involving \( x \) on the other.

Step 2: Analyze the Given Form

The given equation is \( y' = f(y) \). Here, \( y' \) is the derivative of \( y \) with respect to \( x \), and \( f(y) \) is a function of \( y \) alone. Since \( x \) does not explicitly appear in the function \( f(y) \), this suggests it might be possible to express this equation in a separable form.

Step 3: Check for Separability

The equation can be written as \( \frac{dy}{dx} = f(y) \). We can multiply both sides by \( dx \) to obtain \( dy = f(y)dx \). We can then divide both sides by \( f(y) \) (assuming \( f(y) eq 0 \)) to isolate terms: \( \frac{1}{f(y)}dy = dx \). This equation is now in a separable form, with all terms involving \( y \) on one side and terms involving \( x \) on the other.

Step 4: Conclusion on Separability

Since we have successfully manipulated the equation \( y' = f(y) \) into a separable form \( \frac{1}{f(y)}dy = dx \), it demonstrates that every differential equation of the form \( y' = f(y) \) is indeed separable.

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They provide a powerful way to describe various physical phenomena such as motion, heat, and wave propagation. In simple terms, a differential equation tells you how a quantity changes as it moves along. There are different types of differential equations, each designed for specific types of problems.

• Ordinary Differential Equations (ODEs): These involve functions of a single variable and their derivatives. The equation we are considering, \( y' = f(y) \), is an example of an ODE.
• Partial Differential Equations (PDEs): These involve functions of multiple variables and their partial derivatives.

The goal with differential equations is often to find a function or a set of functions that satisfy the equation. Understanding the nature of differential equations is crucial for solving engineering, physics, and mathematics problems effectively.

Separation of Variables

Separation of variables is a straightforward technique used to solve certain differential equations. It is applicable when the equation can be rearranged so that each variable and its differential can be separated onto different sides of the equation. This method simplifies the process of finding a solution by allowing us to integrate each side separately.
To use the separation of variables technique:

• Start by expressing the differential equation in a form where the variables can be separated. For example, \( y' = f(y) \) can be rewritten as \( \frac{dy}{dx} = f(y) \).
• Rearrange the equation to isolate the variables, yielding something like \( g(y)dy = h(x)dx \).
• Integrate both sides to find the solution to the equation.

This method works best when each side of the equation can be expressed solely in terms of one variable—either \( y \) or \( x \). If you can separate the variables, as demonstrated with the equation \( y' = f(y) \), then the equation is described as separable.

Initial Value Problems

Initial value problems (IVPs) are a type of problem involving differential equations where you need to find a specific solution given an initial condition. This initial condition is usually presented as known values of the function and its derivative at a certain point.
Solving an initial value problem typically follows this approach:

• First, solve the differential equation for the general solution. This could involve using separation of variables or other techniques.
• Next, substitute the initial condition into the general solution to find any constants of integration that satisfy the condition.
• The result is a specific solution suited to the initial physical context described by the initial condition.

For example, if you are given the differential equation \( y' = f(y) \) with an initial condition \( y(x_0) = y_0 \), you would first find the general solution by separating variables and then use the initial condition to pinpoint the precise solution. Initial Value Problems are essential in fields like physics and engineering where initial conditions significantly impact the system's behavior.