Solve the differential equation in Exercises 9–22.
19. y²(dy/dx) = 3x²y³ - 6x²

검증된 단계별 안내

Rewrite the given differential equation \(y^{2} \frac{dy}{dx} = 3x^{2} y^{3} - 6x^{2}\) to isolate \(\frac{dy}{dx}\). Divide both sides by \(y^{2}\) (assuming \(y \neq 0\)) to get \(\frac{dy}{dx} = 3x^{2} y - \frac{6x^{2}}{y^{2}}\).

Recognize that the equation is separable or can be manipulated into a separable form. Try to express all terms involving \(y\) on one side and all terms involving \(x\) on the other side.

Rewrite the equation as \(\frac{dy}{dx} = 3x^{2} y - 6x^{2} y^{-2}\). This suggests grouping terms to separate variables: \(\frac{dy}{dx} = x^{2} (3y - 6 y^{-2})\).

Express the differential equation in differential form: \(dy = x^{2} (3y - 6 y^{-2}) dx\). Then rearrange to isolate \(y\) terms with \(dy\) and \(x\) terms with \(dx\): \(\frac{dy}{3y - 6 y^{-2}} = x^{2} dx\).

Integrate both sides: integrate \(\int \frac{dy}{3y - 6 y^{-2}}\) with respect to \(y\) and \(\int x^{2} dx\) with respect to \(x\). After integration, include the constant of integration and solve for \(y\) if possible.

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

Algebraic Manipulation of Differential Equations

Before solving, it is often necessary to rearrange and simplify the differential equation by factoring or dividing terms. This step helps isolate dy/dx and express the equation in a form suitable for applying solution methods like separation of variables.

Integration Techniques

Solving the separated differential equation requires integrating functions of x and y. Familiarity with basic integration rules and techniques, such as power rule and substitution, is essential to find the implicit or explicit solution.

Solve the differential equation in Exercises 9–22.19. y²(dy/dx) = 3x²y³ - 6x²