# 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?