Solve the differential equation in Exercises 9–22.
9. 2√(xy) (dy/dx) = 1, x, y > 0

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Rewrite the given differential equation: \(2\sqrt{xy} \frac{dy}{dx} = 1\) with \(x, y > 0\).

Isolate \(\frac{dy}{dx}\) by dividing both sides by \(2\sqrt{xy}\): \(\frac{dy}{dx} = \frac{1}{2\sqrt{xy}}\).

Express \(\sqrt{xy}\) as \(\sqrt{x} \sqrt{y}\) to separate variables: \(\frac{dy}{dx} = \frac{1}{2 \sqrt{x} \sqrt{y}}\).

Rewrite the equation to separate variables \(y\) and \(x\): multiply both sides by \(2 \sqrt{y}\) and multiply both sides by \(dx\) to get \(2 \sqrt{y} dy = \frac{1}{\sqrt{x}} dx\).

Integrate both sides: \(\int 2 \sqrt{y} \, dy = \int \frac{1}{\sqrt{x}} \, dx\), then solve the resulting integrals to find the implicit or explicit solution for \(y\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Separable Differential Equations

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

Implicit Differentiation and Manipulation

Implicit differentiation involves differentiating both sides of an equation involving x and y without explicitly solving for y. In this problem, rewriting the equation to isolate dy/dx and expressing terms in a form suitable for separation is essential.

Integration Techniques

Solving the separated equation requires integrating functions of x and y. Familiarity with integration methods, including substitution and handling radicals like √(xy), is necessary to find the explicit or implicit solution.

Solve the differential equation in Exercises 9–22.9. 2√(xy) (dy/dx) = 1, x, y > 0