Why e-Integral Factor Omits Constant Integral?

[bennyska]

whenever i see that integrating factor for solving a linear differential equation with
e^{int. p(x) dx} and then multiplied out in the equation, there seems to be no constant. i tried solving an equation with it the other day, and got an incorrect solution because of it (i think. at least i got a correct solution when i neglected it).
why is this?

 

[torquil]

Check this example:

http://en.wikipedia.org/wiki/Examples_of_differential_equations

The section "Non-separable first-order linear ordinary differential equations". An arbitrary additive term in the integral in the exponential, can be written as a constant prefactor on $$\mu$$. Since the whole equation is multiplied by $$\mu$$, this is irrelevant for finding the solution.

And you can se in the explicit expression for the final solution y that you have the freedom to redefine $$\mu$$ by multiplying it ba any nonzero number, since the parameter C is not determined.

Torquil

 

[bennyska]

cool, thanks.