# Substituting variables in integral functional equations

If $$f(x)$$ is differentiable, and equals $$x^2+\int_0^x{e^{-t}f(x-t)}dt$$ , then find $$f(x)$$

Substituting $$x=0$$, we find that $$f(0)=0$$. Then, substituting $$x=t$$ gives us $$f(t)= t^2+\int_0^t{e^{-t}f(0)}dt = t^2$$

However, if we differentiate the given equation, and then solve the resulting differential equation, we get $$f(x)=x^3/3+x^2$$. It can be verified that this is the actual solution to the functional equation.

My doubt is, why does method $$1$$ not work?

• $t$ is a dummy variable and we cannot substitute $x$ by $t$. Jun 15, 2021 at 7:12
• Can you please elaborate? In what situations can substitutions be made, and why isn't it allowed in this case? Jun 15, 2021 at 7:13

When you write $$\int_{0}^{x} e^{-t} f(x-t) dt$$ here the integrating variable is $$t$$, and hence, $$x$$ is to be treated as a constant. Thus the substitution $$t=x$$ is meaningless, as the variable cannot be substituted to be a constant during integration.