In Exercises 1–22, solve the differential equation.
y' = xeʸ√(x-2)

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Rewrite the given differential equation as \(\frac{dy}{dx} = x e^{y} \sqrt{x - 2}\) to clearly identify the variables and their derivatives.

Separate the variables by dividing both sides by \(e^{y}\) and multiplying both sides by \(dx\), giving \(e^{-y} dy = x \sqrt{x - 2} \, dx\).

Integrate both sides separately: integrate \(e^{-y} dy\) with respect to \(y\) on the left side, and integrate \(x \sqrt{x - 2} \, dx\) with respect to \(x\) on the right side.

For the right side integral, consider using a substitution such as \(u = x - 2\) to simplify the integral \(\int x \sqrt{x - 2} \, dx\).

After integrating both sides, include the constant of integration \(C\) and solve for \(y\) if possible to express the general solution.

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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 integration methods, including substitution and handling functions like exponential and square roots, is essential to evaluate the integrals correctly.

Domain Considerations for the Solution

The presence of √(x-2) restricts the domain to x ≥ 2 to keep the expression real. Understanding domain constraints ensures the solution is valid and helps identify any initial conditions or intervals where the solution applies.

In Exercises 1–22, solve the differential equation.y' = xeʸ√(x-2)