The last 3 terms appear to have the same derivatives... combined with their coefficient they sum to zero.
So only the first 2 terms remain.
Those 2 make up the so called laplacian of f.
That will only be zero if you have more information than you have shown.
However, if you treat it as a differential equation that is equal to zero, the solution is:
where g and h are arbitrary functions.
Here is the solution of Wolfram|Alpha.