I came across this problem when I was trying to evaluate $y=\int -a/(bx+c)\mathrm{dx}$, where a, b, and c are constants by using differential equations. By using integration by substitution, I came up with
y=-(a/b)(ln(bx+c)+G) (1)

for some constant G.

But by changing the equation to

dx/dy+bx/a=-c/a

and multiplying by e^by/a , then using the product rule, I got

x=-(c/b)+(D/e^by/a )

So,

y=(a/b)ln(D/[x+(c/b)]) (2)

for some constant D.

(2) doesn't seem to agree with (1), unless D and G are functions of x, which they are not. What is wrong with what I've done?

Brad

Nothing is wrong with what you have done.

Both equations (1) and (2) are correct.

By the time I've posted this I'm sure you will have realised. But anyway...

Firstly:
ln(AB) = ln(A) + ln(B)

ln(A^B ) = Bln(A)

Looking at (1):
-by/a=ln(bx+c) + G
=ln(b(x+(c/b))) + G
=ln(b) + ln(x+(c/b)) + G
=ln(x+(c/b))+(G+ln(b))

Let G+ln(b)=K,
-by/a = ln(x+(c/b)) + K (3)

Looking at (2):
-by/a=-ln(D/(x+(c/b)))
=-ln(D) - ln((x+(c/b))^-1 )
=ln(x+(c/b)) - ln(D)

Let -ln(D) = K, and again we have:
-by/a = ln(x+(c/b)) + K (3)

Therefore (1) and (2) are the same.

Peter