Why no constant of integration for the integration factor?

1. Dec 17, 2011

In applying the usual formula for y'+py=q with u=e∫p and y=(∫uq +C)/u, the results of the indefinite integral to calculate u is without a constant. Why is this? The result is not the same.
For example, suppose I take the problem
dy/dx +y = x*exp(-x). Then, in the usual manner, I get a nice exp(-x)*(x2/2 +C). But if I were to put constants in for the integration factor, I get
exp(-x)*(C1x2 +C2)
which is not the same thing.

2. Dec 17, 2011

JHamm

$$y' + py = q$$
Define $\displaystyle p = P'$
$$u = e^{\int p\ dx}$$
$$= Pe^C \Rightarrow u = AP$$
Then $\displaystyle (yu)' = uq$
$$y = \frac{\int uq\ dx}{u}$$
$$= \frac{\int APq\ dx}{AP}$$
$$= \frac{A}{A}\frac{\int Pq\ dx}{P}$$
So adding the constant doesn't change anything :)

3. Dec 17, 2011