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## Homework Statement

Let M(x,y) = yf(xy) and N(x,y) = xg(xy), where f(v) and g(v) are functions of a single real variable v, defined and continuously differentiable for all real values of v. Under what conditions on f and g is the form Mdx + Ndy exact for all values of x, y in the plane? In that case, find the function u(x,y) such that exact equation holds, and use that information to infer the general solution to the equation y' = -(M(x,y)/N(x,y)).

## The Attempt at a Solution

Okay, so the way I see it, I have to use the product rule and the chain rule to take the derivatives, so

[tex] \frac{\partial M}{\partial y} = xyf_y(xy) + f(xy) [/tex]

and

[tex] \frac{\partial N}{\partial x} = xyg_x(xy) + g(xy) [/tex].

If the form is exact, then we have

[tex] f(xy) + xyf_y(xy) = g(xy) + xyg_x(xy) [/tex]

so maybe a good condition is something like

[tex] \frac{f(xy) - g(xy)}{f_y(xy) - g_x(xy)} = -xy [/tex]

provided the denominator is nonzero. But then, how do use this to find u(x,y)? I need to integrate M(x,y) with respect to x, but f(xy) depends on x. I have no clue what f is, and I don't see how this condition will help at all. What am I missing or doing wrong? Thanks for any help.