In Exercises 1–22, solve the differential equation.

y' = xy ln x ln y

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Rewrite the given differential equation \(y' = xy \ln x \ln y\) in Leibniz notation as \(\frac{dy}{dx} = xy \ln x \ln y\).

Separate the variables by dividing both sides to isolate \(y\) terms on one side and \(x\) terms on the other: write it as \(\frac{1}{y \ln y} dy = x \ln x \, dx\).

Integrate both sides separately: compute \(\int \frac{1}{y \ln y} dy\) on the left and \(\int x \ln x \, dx\) on the right.

For the left integral, use substitution \(u = \ln y\) which implies \(du = \frac{1}{y} dy\), transforming the integral into \(\int \frac{1}{u} du\).

For the right integral, use integration by parts where you let \(u = \ln x\) and \(dv = x \, dx\), then apply the formula \(\int u \, dv = uv - \int v \, du\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Separable Differential Equations

A separable differential equation can be written as a product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the solution.

Properties of Logarithmic Functions

Understanding the natural logarithm function, ln(x), is essential, including its domain (x > 0) and its behavior. Logarithmic properties help simplify expressions and manipulate terms involving ln(x) and ln(y) during the solution process.

Integration Techniques

Solving the separated equation requires integrating functions involving logarithms and polynomials. Familiarity with integration methods, such as substitution and integration by parts, is important to evaluate the integrals correctly.

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