First-Order Linear Equations
Solve the differential equations in Exercises 1–14.

e²ˣy' + 2e²ˣ y = 2x

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Rewrite the given differential equation in the standard linear form \(y' + P(x)y = Q(x)\). Start by dividing the entire equation by \(e^{2x}\) to isolate \(y'\):

\[y' + 2y = 2xe^{-2x}\]

Identify the integrating factor \(\mu(x)\), which is given by \(\mu(x) = e^{\int P(x) \, dx}\). Here, \(P(x) = 2\), so calculate:

\[\mu(x) = e^{\int 2 \, dx} = e^{2x}\]

Multiply both sides of the differential equation by the integrating factor \(e^{2x}\) to make the left side a product derivative:

\[e^{2x} y' + 2 e^{2x} y = 2x\]

Recognize that the left side is the derivative of \(e^{2x} y\), so write:

\[\frac{d}{dx} \left( e^{2x} y \right) = 2x\]

Integrate both sides with respect to \(x\) to find \(e^{2x} y\):

\[e^{2x} y = \int 2x \, dx + C\]

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주요 개념

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

First-Order Linear Differential Equations

These are differential equations of the form y' + P(x)y = Q(x), where y' is the first derivative of y. They can be solved using an integrating factor, which simplifies the equation into an exact derivative, allowing integration to find the solution.

Integrating Factor Method

The integrating factor is a function, usually denoted μ(x), defined as e^(∫P(x)dx). Multiplying the entire differential equation by μ(x) transforms it into a form where the left side is the derivative of μ(x)y, enabling straightforward integration to solve for y.

Exponential Functions and Their Properties

Exponential functions like e^(2x) appear in the equation and integrating factor. Understanding their differentiation and integration properties is essential, as they often simplify the process of finding the integrating factor and solving the differential equation.

First-Order Linear EquationsSolve the differential equations in Exercises 1–14.e²ˣy' + 2e²ˣ y = 2x