Given that $()$ is differentiable at least once,(for all 'x' belongs to set of real numbers) and satisfies the property, $( + ) = () + () $, and $′(0) = 2$. *Show that $′()= () + 2$ and that $() = 2^$
I have tried substituting values for $x$ and even got a relation between $f(2x)$ and $f(x) $ $f(2x)=2(e^x)f(x)$
Yet I am unable to get how should I go about to prove the required.
First, let's calculate $f'(x)$. \begin{align*} f'(x) &= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\ &= \lim_{h \to 0} \frac{(e^{x}f(h) + e^{h}f(x)) - f(x)}{h} \\ &= e^{x}f'(0) + f(x) \\ &=f(x) + 2e^{x}. \end{align*} Here, I use $$ \lim_{h\to 0} \frac{e^{h} - 1}{h} = 1. $$ By solving the differential equation $f'(x) = f(x) + 2e^{x}$ and $f'(0) = 2$, we can obtain $f(x) = 2xe^{x}$.
