Continuous Functions with Constraints: f(x) > -1 and f(x) = x + ∫₁ˣf(t) dt

[erogol]

Find all continuous functions f(x) satisfying
f(x) > −1 and $$f(x) = x + \int_{1}^{x} f(t) dt$$


for all x

 

[lurflurf]

First show f must be twice differentiable.
Next differentiate your equation twice to yield
f''=f'
find f