Problem 71 - Calculus Early Transcendentals

Question

Writing A student objects to Step 2 in the method of separation of variables because one side of the equation is integrated with respect to \(x\) while the other side is integrated with respect to \(y .\) Answer this student's objection. \([\)Hint: Recall the method of integration by substitution. \(]\)

Step-by-Step Solution

Answer

Separate variables using substitution; integrating separately is valid.

Step 1: Understand the Method

The separation of variables for differential equations involves separating the equation into two parts: one dependent on \(x\) and the other dependent on \(y\). Both sides of the equation are then integrated separately.

Step 2: Set Up the Separated Equation

Suppose our differential equation is of the form \( \frac{dy}{dx} = g(y)h(x)\). We rearrange terms to get \( \frac{dy}{g(y)} = h(x)dx \), effectively separating variables between \(y\) and \(x\).

Step 3: Integrate Both Sides

Integrate each side of the equation: \( \int \frac{1}{g(y)} \, dy = \int h(x) \, dx \). Notice that we respect the separation by leaving \(g(y)\) and \(h(x)\) on different sides of the equation.

Step 4: Consider the Substitution Concept

This step involves understanding that integrating with respect to different variables is feasible because each side is effectively a change of variables. Integration by substitution explains how this is valid because differential variables \(dy\) and \(dx\) are consistent within each integral.

Step 5: Resolve the Objection

The objection arises from the concern that variables \(x\) and \(y\) are mixed, but their separation and subsequent integration are consistent and valid due to the method of substitution, where each integral is independently handling its respective variable.

Key Concepts

Integration by Substitution

Integration by substitution is a fundamental technique often used when solving integrals, particularly when dealing with more complex integrals in calculus. It simplifies an integral by considering a change of variables, making it easier to solve.

• Let's say you have an integral that looks complicated in its current form. By introducing a new variable (commonly denoted as \( u \)), which substitutes part of the original integral, you can transform the integral into a simpler form.
• An essential part of this method involves identifying a part of the original integral that can be replaced by a single variable \( u = f(x) \), and then using its derivative \( du = f'(x) \, dx \) to switch the variable of integration from \( x \) to \( u \).
• This process makes the integral easier to evaluate because it often reduces the complexity of the function you need to integrate.

By using substitution, each integral retains consistency, even if different variables are involved. It ensures that each side of the equation, when separated, remains correct and corresponds to the respective variable. This method upholds the validity of solving integrals across different variables while maintaining accuracy through proper substitution.

Differential Equations

Differential equations involve equations that include derivatives, representing rates of change. These are powerful tools for modeling various physical and abstract systems.

• They come in different forms, such as ordinary differential equations (ODEs) which contain functions of a single variable and their derivatives.
• In a basic example, a differential equation might describe how a population changes over time, or how a car's speed varies over a journey.
• Solving differential equations involves finding a function, or set of functions, that satisfies the original equation.

Understanding how to manage these equations is crucial in fields such as engineering, physics, and economics as they provide insights into the dynamic behavior of systems. The process of separation of variables is often applied to solve differential equations, by splitting them into parts that depend on different variables, facilitating a manageable path to a solution.

Variable Separation

Variable separation is a technique used to solve certain differential equations by separating the variables into two parts, each depending on a different variable.

• This method is particularly effective for solving simple equations where terms can be isolated on either side of the equation based on their dependencies.
• In variable separation, you rearrange the terms to ensure that all derivatives and functions of \( y \) are on one side of the equation, and all derivatives and functions of \( x \) are on the other side. An example form is \( \frac{dy}{dx} = g(y)h(x) \).
• Once the variables are appropriately separated, integration can be performed independently on each side of the equation, facilitating the solution.

The separation technique relies on the ability to integrate both sides while respecting the distinct variables \( x \) and \( y \), allowing for a coherent analysis even when individually handling different constants and functions. This powerful technique helps in deriving solutions to equations that model a wide range of real-world phenomena.