I'm given a functional equation $f(x,y)$. I've been told by my teachers to solve such equations by first partially differenciating the equation wrt one variable ,keeping the other variable constant, and then plugging $0,-1,$ etc into one of the variables. However, when I do so in the above problem, I obtain different answers each time, depending on which variable I partially differenciated it in the first step, and which variable I chose to plug in some value$(0)$ in the second step.
I want to know why does this happen? Shouldn't I get the same answer each time? Please guide me
Also,there is a neat way to do the problem,to plug $x=0$ in the original equation itself,we get $f(x)=f(0)-x^2/2 $ So we know for certain that it is the correct answer.
Some context:Yes,I understand that this problem don't require any derivatives,but This question appeared in a test which is very time-bound,and we don't get much time to focus on a particular question.It would be very unlikely that I would be able to notice that plugging x=0 just does the trick.So I just want to solve every such functional equation in a general manner,using partial derivatives,so I dont waste time looking for alternatives in the test.So,requesting you all to solve this by partial derivatives only