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$$\dfrac{dy}{dx} + p(x)y = f(x)$$ 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?

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1 Answer

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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.

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