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

y = 1 + ∫₀ ͯ y(t) dt

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Identify the given integral equation: \(y(x) = 1 + \int_0^x y(t) \, dt\).

Differentiate both sides of the equation with respect to \(x\) to eliminate the integral. Use the Fundamental Theorem of Calculus, which states that \(\frac{d}{dx} \int_0^x y(t) \, dt = y(x)\).

After differentiation, the equation becomes \(\frac{dy}{dx} = y(x)\).

Rewrite the equation as a first-order differential equation: \(y' = y\).

Determine the initial condition by evaluating the original equation at \(x=0\): \(y(0) = 1 + \int_0^0 y(t) \, dt = 1 + 0 = 1\). So, the initial condition is \(y(0) = 1\).

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주요 개념

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

Integral Equations

An integral equation expresses a function in terms of an integral that involves the function itself. Solving such equations often involves converting them into differential equations, which can be easier to analyze and solve using standard calculus techniques.

Differentiation Under the Integral Sign

This technique involves differentiating an integral whose limits or integrand depend on a variable. It allows transforming integral equations into differential equations by applying the Fundamental Theorem of Calculus to differentiate both sides with respect to the variable.

Initial Conditions for Differential Equations

Initial conditions specify the value of the unknown function at a particular point, ensuring a unique solution to a differential equation. When converting integral equations to differential form, identifying the correct initial condition is essential for solving the resulting equation.

Integral EquationsIn Exercises 7–12, write an equivalent first-order differential equation and initial condition for y.y = 1 + ∫₀ ͯ y(t) dt