In Exercises 1–22, solve the differential equation.

y' = (y²-1)x⁻¹

검증된 단계별 안내

Rewrite the given differential equation in a more explicit form: \(y' = \frac{dy}{dx} = \frac{y^{2} - 1}{x}\).

Recognize that this is a separable differential equation because the right side can be expressed as a product of a function of \(y\) and a function of \(x\).

Separate the variables by rewriting the equation as \(\frac{dy}{y^{2} - 1} = \frac{dx}{x}\).

Integrate both sides: integrate \(\int \frac{1}{y^{2} - 1} \, dy\) on the left and \(\int \frac{1}{x} \, dx\) on the right.

After integration, solve the resulting equation for \(y\) implicitly or explicitly, and include the constant of integration.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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 y and a function of x, allowing variables to be separated on opposite sides of the equation. This form enables integration with respect to each variable independently to find the solution.

Integration Techniques

Solving separable equations requires integrating both sides after separation. Familiarity with integrating rational functions, such as partial fraction decomposition, is essential when dealing with expressions like (y² - 1) in the denominator or numerator.

Initial Conditions and General Solutions

After integrating, the solution typically includes an arbitrary constant representing a family of solutions. Applying initial conditions, if given, helps determine the particular solution. Understanding the difference between general and particular solutions is key.

In Exercises 1–22, solve the differential equation.y' = (y²-1)x⁻¹