Answer: for Wisconsinites this should be no problem: it’s just Eau Claire backwards. (“Claire Eau”)
(a)Does there exist a function $f(x, y)$ such that \[ f_x = ax^2y + bxy^2, \quad f_y = y^3+x^3 \] where $a,b$ are constants?
(b) Does there exist a function $f(x, y)$ such that \begin{align*} f_x &= ax^2y + bxy^2+k\cos x, \\ f_y &= y^3+x^3-l e^y \end{align*} where $a, b, k, l$ are constants?
(a) Clairaut requires $f_{xy} = f_{yx}$, which leads to $a=3$, $b=0$. So there is no solution unless $a=3$ and $b=0$. If $a=3$ and $b=0$ then the solution is $f(x,y) = x^3y + \frac14 y^4 $.
(b) Again, there is no solution unless $a=3$ and $b=0$. There is no restriction on $k$ or $l$. If $a=3$ and $b=0$ then the solution is $f(x,y) = x^3y + \frac14 y^4 + k\sin x - le^y$.