In Exercises 1–22, solve the differential equation.

sec x dy + x cos² y dx = 0

검증된 단계별 안내

Rewrite the given differential equation in the form \(M(x,y)\,dx + N(x,y)\,dy = 0\). Here, the equation is \(\sec x \, dy + x \cos^{2} y \, dx = 0\), so rearranged it becomes \(x \cos^{2} y \, dx + \sec x \, dy = 0\).

Check if the differential equation is exact by computing the partial derivatives \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\), where \(M = x \cos^{2} y\) and \(N = \sec x\).

If the equation is not exact, look for an integrating factor that depends on either \(x\) or \(y\) to make it exact. This often involves checking if \(\frac{1}{N} (\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})\) is a function of only one variable.

Once the equation is exact (or made exact), find the potential function \(\Psi(x,y)\) such that \(\frac{\partial \Psi}{\partial x} = M\) and \(\frac{\partial \Psi}{\partial y} = N\) by integrating \(M\) with respect to \(x\) and then determining the function of \(y\).

Write the implicit solution as \(\Psi(x,y) = C\), where \(C\) is a constant, representing the general solution to the differential equation.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Differential Equations

A differential equation relates a function with its derivatives. Solving it means finding the function that satisfies this relationship. In this problem, the equation involves derivatives of y with respect to x, and the goal is to find y as a function of x.

Separable Equations

A separable differential equation can be written so that all terms involving y are on one side and all terms involving x are on the other. This allows integration of each side independently. Recognizing separability is key to solving equations like the one given.

Integration Techniques

Solving separable equations requires integrating functions of x and y separately. Familiarity with integrating trigonometric functions such as sec x and cos² y is essential. Proper integration leads to an implicit or explicit solution of the differential equation.

In Exercises 1–22, solve the differential equation.sec x dy + x cos² y dx = 0