Finding a function from given conditions.

Question

A function $g$ is defined such that $g'(x)$ exists for every real x and satisfies $g'(0)=2$ and $g(x+y)=e^xg(x)+e^yg(y)$ for every $x$ and $y$. What is $g$ ?

I assumed that $g$ is of the form $e^x f(x)$ where $f$ is some function. But I cant make out what is $g$. Is my assumption wrong? What is the right approach?

Note: The wordings of the question when it was initially asked caused some cofusion. I assumed that $g$ was some type of a function involving product of $e^x$ with something else. This is not a given condition.

Answer 1

Put $x=y=0$ and we get $g(0)=g(0)+g(0)$ so $g(0)=0$.

Just put $y=0$ and then $g(x)=e^x g(x)+g(0)=e^x g(x)$. So $g(x)=0$ for all $x$, and then $g'(0)=0$.

Are you sure you have this problem correct?

ADDED IN EDIT

If you meant $g(x+y)=e^yg(x)+e^xg(y)$ then if we set $g(x)=e^x h(x)$ we get $h(x+y)=h(x)+h(y)$. The only continuous solution to this is $h(x)=ax$ for a constant $a$, and so $g(x)=axe^x$. As $g'(0)=2$ then $a=2$.

Answer 2

Putting $x=y=0$ we get $g(0)=0$,just putting $y=0$ we get $$g(x)=e^xg(x)\\g(x)(e^x-1)=0$$ So $g(x)=0$ but $g(x)=0$ doesn't satisfy $g'(0)=2$ so no such function.