Solve the differential equation in Exercises 9–22.
13. (dy/dx) = √y cos²√y

검증된 단계별 안내

Recognize that the given differential equation is \( \frac{dy}{dx} = \sqrt{y} \cos^{2}(\sqrt{y}) \). Our goal is to separate variables to integrate both sides.

Make the substitution \( t = \sqrt{y} \), which implies \( y = t^{2} \) and \( \frac{dy}{dx} = 2t \frac{dt}{dx} \). Rewrite the differential equation in terms of \( t \): \( 2t \frac{dt}{dx} = t \cos^{2}(t) \).

Simplify the equation by dividing both sides by \( t \) (assuming \( t \neq 0 \)): \( 2 \frac{dt}{dx} = \cos^{2}(t) \). Then, solve for \( \frac{dt}{dx} \): \( \frac{dt}{dx} = \frac{1}{2} \cos^{2}(t) \).

Separate variables to isolate \( t \) and \( x \): \( \frac{dt}{\cos^{2}(t)} = \frac{1}{2} dx \). Recall that \( \frac{1}{\cos^{2}(t)} = \sec^{2}(t) \), so the equation becomes \( \sec^{2}(t) dt = \frac{1}{2} dx \).

Integrate both sides: \( \int \sec^{2}(t) dt = \int \frac{1}{2} dx \). Use the integral formula \( \int \sec^{2}(t) dt = \tan(t) + C \) to find the implicit solution, then substitute back \( t = \sqrt{y} \) to express the solution in terms of \( y \) and \( x \).

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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.

Integration Techniques

Solving the equation requires integrating functions involving square roots and trigonometric expressions. Familiarity with substitution methods and standard integral formulas is essential to handle integrals like ∫(1/√y cos²√y) dy or ∫ cos²(u) du.

Chain Rule and Substitution in Integration

Using substitution, such as setting u = √y, simplifies the integral by transforming complex expressions into manageable forms. Understanding how to apply the chain rule in reverse during integration helps in correctly changing variables and integrating composite functions.

Solve the differential equation in Exercises 9–22.13. (dy/dx) = √y cos²√y