Question

If \(\mathrm{f}\) is a continuous function such that \(\int_{0}^{x} f(t) d t=x e^{2 x}+\int_{0}^{x} e^{-t} f(t) d t\) for all \(\mathrm{x}\), find an explicit formula for \(\mathrm{f}(\mathrm{x})\).

Short answer

Question: Find an explicit formula for the continuous function \(f(x)\) that satisfies the equation \(\int_{0}^{x} f(t) d t=x e^{2 x}+\int_{0}^{x} e^{-t} f(t) d t\). Answer: The explicit formula for \(f(x)\) is \(f(x) = \frac{e^{2x}(1+2x)}{1 - e^{-x}}\).

Step-by-step solution

Differentiate Both Sides of the Equation

To find an explicit formula for \(\mathrm{f(x)}\), we'll start by differentiating both sides of the given equation, using linearity of differentiation. By doing this, we hope to isolate \(\mathrm{f(x)}\) on one side of the equation: \(\frac{d}{dx} \left(\int_{0}^{x} f(t) d t\right) = \frac{d}{dx} \left(x e^{2 x}+\int_{0}^{x} e^{-t} f(t) d t\right)\)

Differentiate The Integrals

Now we will apply the Fundamental Theorem of Calculus, which states that if \(\mathrm{F}\) is an antiderivative of \(\mathrm{f}\), then the derivative of the integral of \(\mathrm{f}\) with respect to \(\mathrm{x}\) is just the function \(\mathrm{f}\) itself: \(f(x) = \frac{d}{dx} \left(x e^{2 x}+\int_{0}^{x} e^{-t} f(t) d t\right)\) Next, we differentiate the other side of the equation: \(f(x) = \frac{d}{dx}(x e^{2x}) + \frac{d}{dx} \left(\int_{0}^{x} e^{-t} f(t) d t\right)\) Autoresizing

Apply the Product Rule to Differentiate \(x e^{2x}\)

To differentiate the term \(x e^{2x}\), we need to use the product rule, which states that the derivative of the product of two functions is given by: \((u(x)v(x))' = u'(x)v(x) + u(x)v'(x)\) Where \(u(x) = x\) and \(v(x) = e^{2x}\). Then we compute the derivatives: \(u'(x) = \frac{d}{dx}(x) = 1\) \(v'(x) = \frac{d}{dx}(e^{2x}) = 2e^{2x}\) Now we can apply the product rule: \(\frac{d}{dx}(x e^{2x}) = u'(x)v(x) + u(x)v'(x) = 1\cdot e^{2x} + x\cdot 2e^{2x} = e^{2x}(1+2x)\)

Differentiate the Second Integral Using the Fundamental Theorem of Calculus

Differentiating the second integral: \(\frac{d}{dx} \left(\int_{0}^{x} e^{-t} f(t) d t\right) = e^{-x} f(x)\)

Re-write the Equation and Isolate \(f(x)\)

Now, we can substitute the derivatives back into our original equation: \(f(x) = e^{2x}(1+2x) + e^{-x} f(x)\) To isolate \(f(x)\), we'll subtract \(e^{-x} f(x)\) from both sides of the equation: \(f(x) - e^{-x} f(x) = e^{2x}(1+2x)\) Factor out \(f(x)\): \(f(x)(1 - e^{-x})= e^{2x}(1+2x)\) Finally, solve for \(f(x)\) by dividing both sides by \((1 - e^{-x})\): \(f(x) = \frac{e^{2x}(1+2x)}{1 - e^{-x}}\) Now we have an explicit formula for \(\mathrm{f(x)}\).