Exercise 36

Recall from single variable calculus that, given the derivative of a single variable function and an initial condition, we can integrate to find the original function. We can sometimes use the same process for functions of more than one variable. For example, suppose that a function $f$ satisfies $f_x(x,y) = \cos(y)e^x+2x+y^2$, $f_y(x,y) = -\sin(y)e^x+2xy+3$, and $f(0,0) = 5$.

1. Find all possible functions $f$ of $x$ and $y$ such that $f_x(x,y) = \cos(y)e^x+2x+y^2$. Your function will have both $x$ and $y$ as independent variables and may also contain summands that are functions of $y$ alone.

2. Use the fact that $f_y(x,y) = -\sin(y)e^x+2xy+3$ to determine any unknown non-constant summands in your result from part (a).

3. Complete the problem by determining the specific function $f$ that satisfies the given conditions.