pde – Independence of partial derivatives

I have a HJB set up as follows:

$$ f (x, y) = x + y + f_y (x, y) + f_ {yy} (x, y) $$

Here, I chose a construction so simple that I could probably solve the DE manually. But please, think of the initial problem complicated enough that it is not an option (for example, $ y $ being a vector of several variables).

I want to show that:

$$ f (x) = 1 $$

Intuitively, this should be the case as there is no interaction between $ x $ and $ y $.

Well, formally:

$$
f_x = 1 + 0 + f_ {yx} + f_ {yyx} \
f_y = 1 + f_ {yy} + f_ {yyy} \
f_ {yx} = 0 + f_ {yyx} + f_ {yyyx}
points
$$

I seem to be certain that $ f_ {yx} $ is zero, but just taking derivatives does not get me anywhere. How can I tackle this problem without fully resolving for the DE?