In Exercises 1–22, solve the differential equation.
y' + 3x²y = 7x²

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Identify the type of differential equation. The given equation is a first-order linear differential equation of the form \(y' + P(x)y = Q(x)\), where \(P(x) = 3x^{2}\) and \(Q(x) = 7x^{2}\).

Find the integrating factor (IF), which is given by \(\mu(x) = e^{\int P(x) \, dx}\). Calculate \(\int 3x^{2} \, dx\) to determine the integrating factor.

Multiply both sides of the differential equation by the integrating factor \(\mu(x)\) to make the left side an exact derivative of the product \(\mu(x)y\).

Rewrite the left side as \(\frac{d}{dx}[\mu(x)y]\) and set it equal to the right side multiplied by \(\mu(x)\). This gives \(\frac{d}{dx}[\mu(x)y] = \mu(x) Q(x)\).

Integrate both sides with respect to \(x\) to find \(\mu(x)y = \int \mu(x) Q(x) \, dx + C\). Finally, solve for \(y\) by dividing both sides by \(\mu(x)\).

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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 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 typically e^(∫P(x)dx). Multiplying the entire differential equation by this factor transforms the left side into the derivative of (integrating factor × y), enabling straightforward integration to solve for y.

Integration of Functions Involving Polynomials

Solving the equation requires integrating expressions involving polynomials like x². Understanding basic integration rules for powers of x is essential to compute the integrating factor and the integral of the right-hand side.

In Exercises 1–22, solve the differential equation.y' + 3x²y = 7x²