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

(1+x)y' + y = √x

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Rewrite the given differential equation \((1+x)y' + y = \sqrt{x}\) in the standard linear form \(y' + P(x)y = Q(x)\) by dividing both sides by \((1+x)\), resulting in \(y' + \frac{1}{1+x}y = \frac{\sqrt{x}}{1+x}\).

Identify the integrating factor \(\mu(x)\), which is given by \(\mu(x) = e^{\int P(x)\,dx}\). Here, \(P(x) = \frac{1}{1+x}\), so compute \(\mu(x) = e^{\int \frac{1}{1+x} dx}\).

Calculate the integral \(\int \frac{1}{1+x} dx\), which is \(\ln|1+x|\), and thus the integrating factor becomes \(\mu(x) = e^{\ln|1+x|} = |1+x|\). Since \(1+x > 0\) for the domain of interest, we can write \(\mu(x) = 1+x\).

Multiply the entire differential equation by the integrating factor \(\mu(x) = 1+x\) to get \((1+x)y' + y = \sqrt{x}\) multiplied by \(1+x\), which simplifies back to the left side being the derivative of \(\mu(x)y\), i.e., \(\frac{d}{dx}[(1+x)y] = (1+x) \cdot \frac{\sqrt{x}}{1+x} = \sqrt{x}\).

Integrate both sides with respect to \(x\): \(\int \frac{d}{dx}[(1+x)y] dx = \int \sqrt{x} dx\). This gives \((1+x)y = \int x^{1/2} dx + C\). Then solve for \(y\) by dividing both sides by \((1+x)\).

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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). It involves the first derivative of the unknown function and can be solved using an integrating factor. Recognizing this form is essential to apply the correct solution method.

Integrating Factor Method

The integrating factor is a function, usually denoted μ(x), used to multiply both sides of a linear differential equation to make the left side an exact derivative. It is found by μ(x) = e^(∫P(x)dx), enabling straightforward integration to solve for y.

Solving for y after Integration

After multiplying by the integrating factor, the equation becomes d/dx[μ(x)y] = μ(x)Q(x). Integrating both sides with respect to x and then isolating y gives the general solution. This step requires careful integration and algebraic manipulation.

First-Order Linear EquationsSolve the differential equations in Exercises 1–14.(1+x)y' + y = √x