General Integration Question

1. Jul 26, 2008

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 lets e^C = A (why?).

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

2. Jul 26, 2008

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.

3. Jul 27, 2008

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?

4. Jul 27, 2008

tiny-tim

Hi Oneiromancy!
yes … except

i] it's A = eC1-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.

5. Jul 27, 2008

HallsofIvy

Staff Emeritus
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= eC.