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In a fluid dynamics problem appears a stream function $f(x,y)$ which is defined by the following system
$$\begin{align} &\frac{\partial f}{\partial x}=\frac{g'(x)f(x,y)-a_1}{1-g(x)} \\ &\frac{\partial f}{\partial y}=\frac{a_2}{1-g(x)} \end{align}$$
where $a_1$ and $a_2$ are constants and $g(x)$ some arbitrary function of $x$. I would like to find a general procedure in order to express $f(x,y)$ in terms of the arbitrary function $g(x)$ and the variable $y$, i.e. $f(x,y)=F(g(x),y)$.
HINT
Following the hint in the comments, we note the 2nd equation's RHS does not depend on $y$. We integrate both sides to get $$ f(x,y) = \frac{a_2 y}{1-g(x)} + C(x) $$ where $C(x)$ is some function independent of $y$, arising from the constant of integration. Can you take it from here (compute $f_x$ and compare to the first equation)?
First take a look at these steps
$$\eqalign{ {{\partial f} \over {\partial y}} &= {{{a_2}} \over {1 - g(x)}} \cr f &= {{{a_2}} \over {1 - g(x)}}y + h(x) \cr {{\partial f} \over {\partial x}} &= {{{a_2}g'(x)} \over {{{\left[ {1 - g(x)} \right]}^2}}}y + h'(x) \cr &= {{g'(x)f(x,y) - {a_1}} \over {1 - g(x)}} \cr} $$
and the comparison of the last two equations yields
$${{g'(x)} \over {1 - g(x)}}f(x,y) = {{{a_1}} \over {1 - g(x)}} + {{{a_2}g'(x)} \over {{{\left[ {1 - g(x)} \right]}^2}}}y + h'(x)$$
and finally
$$f(x,y) = {{{a_1}} \over {g'(x)}} + {{{a_2}} \over { {1 - g(x)} }}y + {{1 - g(x)} \over {g'(x)}}h'(x)$$
