Is there a better way to do this? (1 Viewer)

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StatusX

Homework Helper
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I have the pair of equations:

[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?
 

Tide

Science Advisor
Homework Helper
3,078
0
You might try writing

[tex]b = b_0 - \int a dx[/tex]

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.
 

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