The Unknown Function Given: $$\int_0^x f(t)dt + xf(x) = x^2$$ Solution: Let $F(x) = \int_0^x f(t)dt$. Then, the given equation can be written as $$F(x) + xf(x) = x^2$$ Differentiating both sides with respect to $x$ yields $$f(x) + F'(x) + xf'(x) = 2x$$ Substituting $F'(x)$ with $f(x)$ gives $f(x) + xf'(x) = 2x$$ Rearranging […]
