Consider a function $f(x, y) = Ax^5 + Bx^4y + Cx^3y^2 + Dx^2y^3 + xy^4 − y^5$ 
Consider a function $f(x, y) = Ax^5 + Bx^4y + Cx^3y^2 + Dx^2y^3 + xy^4 − y^5$
where $A$, $B$, $C$, $D$ are unspecified real numbers. Determine the values of
$A$, $B$, $C$, $D$ such that $f(x,y)$ satisfies $f_{xx}(x,y) + f_{yy}(x,y) = 0$

What I did so far, I took double derivative of x and y:

$f_{xx}(x,y) = 20Ax^3+12Bx^2y+6Cxy^2+2Dy^3$
$f_{yy}(x,y) = 2Cx^3+6Dx^2y+12xy^2-20y^3$

How can I satisfies $f_{xx}(x,y) + f_{yy}(x,y) = 0$? There are no terms that cancels, subtracts or add.

Add up and you get
$[20A+2C]x^3 + [12B+6D]x^2y + [6C+12]xy^2 + [2D-20]y^3=0$
So
$D = 10$
$C = -2$
$B = -5$
$A = \frac{1}{5}$
