I have 2 functions $$\mu_1(N,H) = aNH+c$$ $$\mu_2(N,H) = aN+bH+c$$
From my (limited) understanding of differential equations I can then write
$$d\mu=\frac{\partial \mu}{\partial N}dN + \frac{\partial \mu}{\partial H}dH$$ and so $$d\mu_1=aHdN+aNdH$$ $$d\mu_2=adN+bdH$$
I think the above is right but if not then that answers my question. How do I get the original functions from these differential equations?
I doesn't seem to be as simple as $$\mu=\int \frac{\partial \mu}{\partial N}dN+\int \frac{\partial \mu}{\partial H}dH$$ Thanks seems to work for $\mu_2$ but not $\mu_1$
Another wrong approach is to note divide by $dN$ and integrate from there $$\frac{du}{dN}=\frac{\partial \mu}{\partial N}+\frac{\partial \mu}{\partial H}\frac{dH}{dN}$$ in both cases $dH/dN=0$ so then integrate with respect to $N$ and then $$\mu=\int \frac{\partial \mu}{\partial N} dN$$ which works for $\mu_1$ but not $\mu_2$
It was very many years ago that I lightly studied this stuff so an intuitive explanation of the general approach will be appreciated.