# 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?

• Try using $e^{a+b}\equiv e^a\cdot e^b$, and now multiply your ODE by your integrating factor. See why we don't need the constant of integration now? Jul 27, 2017 at 7:48
• The explanation is that we're multiplying the equation by some factor so we can apply the reverse product rule. We can choose any factor we want as long as it works, and you should see (when deriving the integrating factor method) that you'll end up with the same result Jul 27, 2017 at 8:52

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.