In Exercises 1–22, solve the differential equation.

y' = sin³ x cos² y

검증된 단계별 안내

Recognize that the given differential equation is separable since it can be written as a product of a function of x and a function of y: \(y' = \sin^{3} x \cos^{2} y\).

Rewrite the differential equation in differential form: \(\frac{dy}{dx} = \sin^{3} x \cos^{2} y\).

Separate the variables by dividing both sides by \(\cos^{2} y\) and multiplying both sides by \(dx\): \(\frac{dy}{\cos^{2} y} = \sin^{3} x \, dx\).

Express the left side in terms of \(\sec^{2} y\) to facilitate integration: \(\sec^{2} y \, dy = \sin^{3} x \, dx\).

Integrate both sides separately: \(\int \sec^{2} y \, dy = \int \sin^{3} x \, dx\), then solve each integral using appropriate techniques (such as substitution or trigonometric identities).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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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 of Trigonometric Functions

Solving the equation involves integrating powers of sine and cosine functions. Familiarity with trigonometric identities and integration techniques, such as substitution or power-reduction formulas, is essential to evaluate these integrals correctly.

Implicit Solutions and Initial Conditions

After integration, solutions may be implicit, involving both x and y in an equation. Understanding how to interpret implicit solutions and apply initial conditions, if given, helps in finding explicit solutions or particular solutions to the differential equation.

In Exercises 1–22, solve the differential equation.y' = sin³ x cos² y