Systems of partial differential equations and the Frobenius theorem

Question

I am trying to solve this. I have completed i and ii.

i is a result of the Frobenius theorem, and ii requires dp/dx=dp/dy.

I am not sure how to solve the system of partial differential equations in iii.

Answer 1

One way is just to compute the path integral. I am not sure I am using the correct terminology, but what I mean is as follows: for fixed $ (x,y) $, let $ ( t x, t y) $ for $ t \in [0,1] $ be a path from the origin to $ (x,y) $. Then $ u(t) = u(x(t), y(t)) $ satisfies the first order equation: \begin{align} \frac{du}{dt} &= \frac{\partial u}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial t} \\ &= f x + g y \\ &= (x+ 2 x y t + y )( 1 + u^2 ) \end{align} which is seperable: $$ \int_0^{u(x,y)} \frac{du}{1+ u^2} = \int_0^1 x+ 2 x y t + y \; dt $$ to find $ u(x,y) = \tan( x + x y + y) $.