Let $f(x)$ be a continuous function and 'c' is a constant satisfying $\int_0^x f(t) dt = e^x - ce^{2x} \int_0^x f(t)^2 dt$, then:
Step-by-Step Solution
Key Concept: Differentiate the functional equation $\int_0^x f(t) dt = e^x - ce^{2x} \int_0^x f(t)^2 dt$ with respect to $x$ to obtain a differential equation for $f(x)$, then use the boundary condition at $x=0$ to determine the constant $c$ through evaluation of an integral.
Substitute $x = 0$ into $0 = 1 - c\int_0^t f(t)e^{-t}dt$ to get $c = \frac{1}{k}$ where $k = \int_0^1 f(t)e^{-t}dt$. Differentiate the given equation to obtain $f'(x) = e^x - 2e^{2x}$. Calculate $k = \int_0^1 (e^t - 2e^{2t})e^{-t}dt = \int_0^1 (1 - 2e^t)dt = 3 - 2e$. Therefore $c = \frac{1}{3-2e}$.
Correct Answer: 2,3