Question

Converting to an exact differential. Given the expression \(d x+(x / y) d y\), show that dividing by \(x\) results in an exact differential. What is the function \(f(x, y)\) such that \(d f\) is \(d x+(x / y) d y\) divided by \(x\) ?

Short answer

Divide by \(x\) to get \( \frac{dx}{x} + \frac{dy}{y}\). Thus, \(f(x, y) = \ln|xy| + C\).

Step-by-step solution

Understand the given differential

The given differential expression is \(dx + \frac{x}{y} dy\).

Divide the differential by x

Divide each term of the differential equation by \(x\): \(\frac{dx}{x} + \frac{\frac{x}{y} dy}{x} = \frac{dx}{x} + \frac{dy}{y}\).

Identify resulting differential

The resulting differential expression is \(\frac{dx}{x} + \frac{dy}{y}\).

Integrate each term

Integrate each term separately: \(\int \frac{dx}{x} + \int \frac{dy}{y} = \ln|x| + \ln|y| + C\).

Combine results

Combine the integrated results into a single function: \(f(x, y) = \ln|x| + \ln|y| + C\).

Simplify the function

Since \(\ln|x| + \ln|y| = \ln|xy|\), the simplified function can be written as \(f(x, y) = \ln|xy| + C\).

Key concepts

Differential Equations

Differential equations describe how a particular quantity changes in relation to another quantity. In this exercise, you are dealing with a first-order differential equation of the form \(dx + \frac{x}{y}dy\). The goal is to manipulate it into an exact differential. An exact differential means that it can be written as the differential of some function \(f(x, y)\). This makes solving the equation easier since integrating the exact differential will give you the function \(f(x, y)\). Understanding differential equations is key in fields like physics, engineering, and even economics where rates of change and slopes are crucial.

Integration

In the context of this solution, integration is used to find the function \(f(x, y)\) whose differential is \(dx + \frac{x}{y} dy\) divided by \(x\). We divide each term by \(x\) and then integrate. This results in:

• \(\frac{dx}{x}\) integrates to \(\text{ln}|x|\)
• \(\frac{dy}{y}\) integrates to \(\text{ln}|y|\)

Combining these integrations gives us \(f(x, y) = \text{ln}|x| + \text{ln}|y| + C\). Since the natural log function has the property that \(\text{ln}|x| + \text{ln}|y| = \text{ln}|xy|\), we can simplify it further to \(f(x, y) = \text{ln}|xy| + C\).

Functions of Multiple Variables

Functions of multiple variables are functions that depend on more than one input. In our solved exercise, \(f(x, y)\) is a function of both \(x\) and \(y\). Understanding these types of functions is critical, especially when dealing with real-world problems like optimizing costs or resources that depend on multiple factors. A function of multiple variables can have partial derivatives, which are derivatives taken with respect to one variable while keeping the others constant. For example, the partial derivative of \(\text{ln}|xy|\) with respect to \(x\) would be \( \frac{1}{x}\) and with respect to \(y\) would be \( \frac{1}{y}\).

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. In mathematics, the natural logarithm \(\text{ln}|x|\) is particularly important because it is the inverse of the natural exponential function \(e^x\). In our exercise, we use the properties of logarithms to integrate the given expressions. Some important properties of logarithms include:

• \(\text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b) \)
• \(\text{ln}(\frac{a}{b}) = \text{ln}(a) - \text{ln}(b)\)
• \(\text{ln}(a^b) = b \times \text{ln}(a) \)

These properties allow us to combine and simplify the logarithmic terms easily, making integration straightforward.