Where ,
$$F(x,y,y')=y'+y$$ for admissible function $y$. Then what is the extremal.
I did the solution using the special case of Euler-Lagrange equation where the $x$ is missing and arrived at the result $y(x)=c$ is the extremal. But while checking the answer in the book, it says that it has no extremal. I am confused. Help me out.
You can check your answer by plugging it in the functional with a perturbation, $y_p=c+\epsilon f(x)$ and seeing if you get a minimum at $\epsilon =0$ for all $f(x)$. $$J(y_p)=\int_a^b (\epsilon f'(x)+c+ \epsilon f(x)) dx $$ Taking a derivative with respect to $\epsilon$ and setting equal to zero yields $$\int_a^b f'(x)+f(x)dx$$ which is not zero for all possible f(x).
