\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) A certain function $f(x,y)$ is
known to have partial derivatives of the form

$${\partial f\over\partial x} = 2xy +g(y),\qquad
{\partial f\over\partial y} = x^2 + 3x.\eqno(*)$$
 
\noindent Note that $g$ is a function of $y$ only.  Use the equality
of mixed partial derivatives (Clairaut's Theorem) to find the function
$g$ up to an arbitrary additive constant.  Then find all functions $f$
with partial derivatives of the form $(*)$.
 
\medskip

\noindent b) Find all functions $f(x,y)$ satisfying
 
 $${\partial f\over\partial x} = 2e^{2x}y + xy^2 + g(y),\qquad
    {\partial f\over\partial y} = x^2y + 4y^3x + h(x).$$
 
\noindent {\bf Hint} The solution contains {\it four}\/ arbitrary
constants.









\vfil\eject\end

