Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

검증된 단계별 안내

Start with the given integral equation: \(y = \ln x + \int_{x}^{e} \sqrt{t^{2} + (y(t))^{2}} \, dt\).

Recognize that the integral has a variable lower limit \(x\) and a constant upper limit \(e\). To differentiate both sides with respect to \(x\), apply the Leibniz rule for differentiation under the integral sign.

Differentiate the right side with respect to \(x\): the derivative of \(\ln x\) is \(\frac{1}{x}\), and by the Leibniz rule, the derivative of the integral \(\int_{x}^{e} f(t) \, dt\) with respect to \(x\) is \(-f(x)\) because the lower limit is \(x\).

Set up the differential equation by differentiating both sides: \(\frac{dy}{dx} = \frac{1}{x} - \sqrt{x^{2} + (y)^{2}}\).

Use the original equation to find the initial condition by substituting \(x = e\): \(y(e) = \ln e + \int_{e}^{e} \sqrt{t^{2} + (y(t))^{2}} \, dt\), which simplifies to \(y(e) = 1\) since the integral from \(e\) to \(e\) is zero.

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주요 개념

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

Integral Equations and Their Relation to Differential Equations

An integral equation expresses a function in terms of an integral involving the function itself. Converting such an equation into a differential equation involves differentiating both sides to eliminate the integral, allowing the use of differential equation techniques to find solutions.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that differentiation can undo integration. When differentiating an integral with a variable limit, the derivative equals the integrand evaluated at that limit, which is essential for transforming the given integral equation into a differential equation.

Initial Conditions in Differential Equations

Initial conditions specify the value of the unknown function at a particular point, ensuring a unique solution to a differential equation. In this problem, identifying the initial condition from the integral equation or given values is crucial for solving the resulting first-order differential equation.

Integral EquationsIn Exercises 7–12, write an equivalent first-order differential equation and initial condition for y.y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt