In Exercises 1–22, solve the differential equation.
y'/2 + y = e⁻ˣ sin x

검증된 단계별 안내

Rewrite the given differential equation to isolate \( y' \). Starting with \( \frac{y'}{2} + y = e^{-x} \sin x \), multiply both sides by 2 to get \( y' + 2y = 2 e^{-x} \sin x \).

Recognize that this is a first-order linear differential equation of the form \( y' + P(x) y = Q(x) \), where \( P(x) = 2 \) and \( Q(x) = 2 e^{-x} \sin x \).

Find the integrating factor \( \mu(x) \) using the formula \( \mu(x) = e^{\int P(x) \, dx} = e^{\int 2 \, dx} = e^{2x} \).

Multiply the entire differential equation by the integrating factor \( e^{2x} \) to obtain \( e^{2x} y' + 2 e^{2x} y = 2 e^{2x} e^{-x} \sin x \), which simplifies to \( \frac{d}{dx} \left( e^{2x} y \right) = 2 e^{x} \sin x \).

Integrate both sides with respect to \( x \): \( \int \frac{d}{dx} \left( e^{2x} y \right) dx = \int 2 e^{x} \sin x \, dx \). This gives \( e^{2x} y = \int 2 e^{x} \sin x \, dx + C \), where \( C \) is the constant of integration.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

First-Order Linear Differential Equations

A first-order linear differential equation has the form y' + P(x)y = Q(x). Solving it involves finding an integrating factor to simplify the equation into an exact derivative, allowing integration to find the general solution.

Integrating Factor Method

The integrating factor is a function, usually e^(∫P(x)dx), used to multiply both sides of a linear differential equation. This transforms the left side into the derivative of a product, making the equation easier to integrate and solve.

Solving Nonhomogeneous Equations with Exponential and Trigonometric Terms

When the nonhomogeneous term involves expressions like e^(-x)sin(x), solutions often require using the method of undetermined coefficients or variation of parameters. Recognizing the form helps in guessing a particular solution that fits the right-hand side.

In Exercises 1–22, solve the differential equation.y'/2 + y = e⁻ˣ sin x