Why does the constant in integration get multiplied instead of canceled out?

[Oneiromancy]

I solved a pretty routine first-order diff. eq. where you simply separate the variables.

xcos(x)(dy/dx) - sin(y) = 0

=> $$\int cot(y)dy$$ = $$\int dx/x$$

Now, I thought that you would get an arbitrary constant, C, on both sides and they would cancel each other out, but that's wrong. My book let's e^C = A (why?).

The answer should be sin(y) = Ax, but I didn't get that because I canceled out the constant. I suppose my question is why does this happen?

 

[Defennder]

The constant C is arbitrary, which means to say it doesn't have a fixed unknown value. Therefore you can't assume that they have the same value on both sides and it cancel each other out.

 

[Oneiromancy]

So I could set the LHS constant to be C_1 and the RHS to be C_2 then their difference can be a new constant A?

 

[tiny-tim]

Hi Oneiromancy!

So I could set the LHS constant to be C_1 and the RHS to be C_2 then their difference can be a new constant A?

yes … except

i] it's A = e^C1^-C2

ii] the examiners will expect you to take the short-cut, and just write one C, on one side of the equation, rather than write two and subtract.

 

[HallsofIvy]

The reason the constant is multiplied is that direction integration
gives you ln(sin(y))= ln(x)+ C and then taking the exponential of both sides,
$$e^{ln(sin(y))}= e^{ln(x)+ C}$$
[tex]sin(y)= e^{ln(x)}e^C= Ax[/itex]
where A= e^C.