In Exercises 1–22, solve the differential equation.

y' = xeˣ⁻ʸ csc y

검증된 단계별 안내

Rewrite the given differential equation \(y' = xe^{x - y} \csc y\) as \(\frac{dy}{dx} = xe^{x - y} \csc y\) to clearly identify the derivative form.

Express the right side to separate variables involving \(y\) and \(x\). Notice that \(e^{x - y} = e^x \cdot e^{-y}\), so rewrite the equation as \(\frac{dy}{dx} = x e^x e^{-y} \csc y\).

Rewrite \(\csc y\) as \(\frac{1}{\sin y}\) and combine it with \(e^{-y}\) to get \(e^{-y} \csc y = \frac{e^{-y}}{\sin y}\). Then, rearrange the equation to isolate \(dy\) and \(dx\) terms: \(\sin y e^y dy = x e^x dx\).

Separate variables by multiplying both sides appropriately to get \(\sin y e^y dy = x e^x dx\), which allows integration on both sides with respect to their own variables.

Integrate both sides: compute \(\int \sin y e^y dy\) on the left and \(\int x e^x dx\) on the right. After integration, include the constant of integration and express the implicit solution.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

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 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 functions of x and y separately. Familiarity with integrating exponential functions, trigonometric functions like cosecant, and their combinations is essential to find the implicit or explicit solution.

Implicit vs. Explicit Solutions

After integration, solutions may be implicit (involving both x and y) or explicit (solved for y). Understanding how to interpret and manipulate implicit solutions is important, especially when the equation involves transcendental functions like csc(y).

In Exercises 1–22, solve the differential equation.y' = xeˣ⁻ʸ csc y