Integral or differential equation?

Calculus Level 3

$f(x)= \int_{0}^{x}e^{x-y}f'(y)dy - (x^2-x+1)e^x$

Find the value of $x$ for which $f(x)=0$ .

( $f'(x)$ denotes the first derivative of $f(x)$ with respect to $x$ .)

1 solution

Brian Moehring
Oct 28, 2018

Multiply both sides of the given equation by $e^{-x}\text{:}$ $e^{-x} f(x) = \int_0^x e^{-y} f'(y)\,dy - (x^2 - x + 1)$

Differentiate both sides with respect to $x\text{:}$ $e^{-x} f'(x) - e^{-x} f(x) = e^{-x} f'(x) - (2x - 1)$

Solve for $f(x)\text{:}$ $f(x) = (2x-1)e^x$ which equals zero when $2x-1 = 0 \iff x=\boxed{\frac{1}{2}}$