[tex]a+\frac{db}{dx}=0[/tex]

[tex]ac+\frac{d(bc)}{dx}=d[/tex]

where a,b,c,d are functions of x. I want to solve for a in terms of c and d. I can do it as follows. Start with the second equation:

[tex]ac+\frac{d(bc)}{dx}=ac+c\frac{db}{dx}+b\frac{dc}{dx}=d[/tex]

Now plug in the first equation:

[tex]ac+c(-a)+b\frac{dc}{dx}=b\frac{dc}{dx}=d[/tex]

Now we have an expression for b, so using the first equation

*again*:

[tex]a=-\frac{d}{dx}\left( \frac{d}{dc/dx} \right) [/tex]

My problem is with using the first equation twice. It seems redundant, but I can't find another way to do it. Can anyone think of a better way, or maybe point out why there isn't one?