Another Functional Equation

Calculus Level pending

Let $f : \mathbb{R} \rightarrow \mathbb{R^+}$ be a function with the following property - $f(x)=\int_{0}^{x} \left[t-f(t)\right] \ \mathrm{d}t$

If $f(0)=0$ , then find $\lfloor 100f(1) \rfloor$

1 solution

Tom Engelsman
Dec 12, 2015

Differentiating the above functional equation with respect to x gives:

f'(x) = x - f(x); f(0) = 0

The above first-order ODE has exp(x) as its integrating factor, thus:

[f(x) exp(x)] ' = x exp(x) (I);

and integrating (i) yields:

f(x) = (x - 1) + C*exp(-x) => which the boundary condition f(0) = 0 yields C = 1. Hence:

f(x) = (x - 1) + exp(-x)

and the floor function of 100*f(1) = 36.