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

검증된 단계별 안내

Recognize that the given differential equation \( \frac{dy}{dx} = 2x(y - 1) \) is separable because the right-hand side can be expressed as a product of a function of \( x \) and a function of \( y \).

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

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

After integration, express the result as \( \ln|y - 1| = x^2 + C \), where \( C \) is the constant of integration.

Since the problem states \( y > 1 \), you can drop the absolute value and solve for \( y \) by exponentiating both sides to get \( y - 1 = e^{x^2 + C} \), then rewrite as \( y = 1 + Ce^{x^2} \) where \( C = e^C \) is a new constant.

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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 separable equations requires integrating both sides after separation. Familiarity with basic integration rules and methods, such as substitution or recognizing standard integral forms, is essential to find the explicit solution.

Initial Conditions and Domain Restrictions

The problem specifies y > 1, which restricts the solution domain. Understanding how initial conditions or domain constraints affect the solution ensures the correct branch of the solution is chosen and the solution is valid within the given context.

Solve the differential equation in Exercises 9–22.17. (dy/dx) = 2x(y - 1), y > 1