Solve the differential equation in Exercises 9–22.
10. (dy/dx) = x²√y, y > 0

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Recognize that the given differential equation is separable. The equation is \(\frac{dy}{dx} = x^{2} \sqrt{y}\), where \(y > 0\).

Rewrite the equation to separate variables \(y\) and \(x\). Express it as \(\frac{dy}{\sqrt{y}} = x^{2} dx\).

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

After integrating, you will get an expression involving \(y\) and \(x\) plus an integration constant \(C\). Write this implicit solution.

Finally, solve the implicit equation for \(y\) if possible, to express \(y\) explicitly as a function of \(x\).

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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 Power Functions

Integrating power functions involves applying the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. This is essential when integrating terms like x² or y raised to a fractional power.

Initial Conditions and Domain Restrictions

The condition y > 0 restricts the solution to positive values of y, which affects the domain and the form of the solution. Understanding initial or domain conditions ensures the solution is valid and meaningful within the given constraints.

Solve the differential equation in Exercises 9–22.10. (dy/dx) = x²√y, y > 0