Why is the constant of integration of the integrating factor irrelevant when solving first-order linear differential equations?

Question

When solving a first order linear differential equation, why don't we add a constant to the integrating factor?

Won't that affect the solution of the equation?

Answer 1

For the equation $y'+P(x)y=Q(x)$ the integrating factor is $e^{\int P(x) dx}$. If we added a constant of integration the integrating factor would become $Ae^{\int P(x)dx}$ where $A=e^{c}\neq0$. Proceeding to solve with our new integrating factor we have $Ae^{\int P(x)dx}y'+AP(x)ye^{\int P(x)dx}=AQ(x)e^{\int P(x)dx}$. This is $\frac{d}{dx}Aye^{\int P(x)dx}=AQ(x)e^{\int P(x)dx}$. Hence $Aye^{\int P(x)dx}=A\int Q(x)e^{\int P(x)dx}$ dx. We now have our constant $A$ on both sides of the equation so we can cancel it since it is nonzero.