Question

For Exercises 127 and 128 find a function f satisfying each equation.128. f(x) = e² + ∫₁ˣ f(t) dt

Accepted Answer

Start by recognizing that the given equation defines the function $$ f(x)  in $$terms of an integral involving itself: $$ f(x) = e^{2} + \int_{1}^{x} f(t) \, dt . $$This is an integral equation that can be converted into a differential equation by differentiation. Differentiate both sides of the equation with respect to $$ x . $$Using the Fundamental Theorem of Calculus, the derivative of the integral $$ \int_{1}^{x} f(t) \, dt $$ with respect to $$ x  is$$ $$ f(x) . So$$, differentiating gives: $$ f'(x) = 0 + f(x) $$, since $$ e^{2}  is a $$constant. Rewrite the resulting differential equation: $$ f'(x) = f(x) . $$This is a first-order linear differential equation whose general solution is well-known. Solve the differential equation $$ f'(x) = f(x) . $$The general solution has the form $$ f(x) = Ce^{x} $$, where $$ C  is a $$constant to be determined. Use the original equation to find the constant $$ C . $$Substitute $$ f(x) = Ce^{x} $$ back into the integral equation and solve for $$ C  by $$evaluating at $$ x = 1  or by $$matching both sides of the equation.

For Exercises 127 and 128 find a function f satisfying each equation.
128. f(x) = e² + ∫₁ˣ f(t) dt

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Key Concepts

Integral Equations

Fundamental Theorem of Calculus

Solving First-Order Differential Equations