Ignoring the constant of integration $C$ in the integrating factor method for solving Linear ODE
$$\dfrac{dy}{dx} + p(x)y = f(x)$$ Solving the linear dif. equation, we can use integrating factor method.
We know integrating factor: $exp(\int p(x) dt) = exp(P(x) + C)$.
But we ignore the constant of integration $C$. How can we explain why the constant was ignored?
Solution 1:
The integrating factor is $\exp(P(x)+C)= K\exp(P(x)),$ where $K=\exp(C) > 0$.
Multiplying to the equation $$K\exp(P(x))(\dfrac{dy}{dx} + p(x)y) = K\exp(P(x))f(x)$$
$$K\frac{d}{dx}(\exp(P(x)) y)=K\exp(P(x))f(x)$$
We can always divide by $K$ anyway, hence there isn't a need to include the constant.