Is there a better way to do this?

[StatusX]

I have the pair of equations:

$$a+\frac{db}{dx}=0$$

$$ac+\frac{d(bc)}{dx}=d$$

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:

$$ac+\frac{d(bc)}{dx}=ac+c\frac{db}{dx}+b\frac{dc}{dx}=d$$

Now plug in the first equation:

$$ac+c(-a)+b\frac{dc}{dx}=b\frac{dc}{dx}=d$$

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

$$a=-\frac{d}{dx}\left( \frac{d}{dc/dx} \right)$$

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?

 

[Tide]

You might try writing

$$b = b_0 - \int a dx$$

from the the first equation and substitute into the second equation. You can ultimately write this as a differential equation for a having eliminated b.