Defining formulas for first-order linear differential equations.

Question

When defining the formulas for the first-order linear differentiable functions we are necessitated to define a equation that satisfies $u'(x)$ = $u(x)p(x)$ so then the product rule can be applied. And as may be noted, the resultant equation for $u(x)$ is can be well defined through the following steps.

\begin{aligned} \frac{\mu^{\prime}(x)}{\mu(x)} & =p(x) \\ \int \frac{\mu^{\prime}(x)}{\mu(x)} d x & =\int p(x) d x \\ \ln |\mu(x)| & =\int p(x) d x+C \\ e^{\ln |\mu(x)|} & =e^{\int p(x) d x+C} \\ |\mu(x)| & =C_1 e^{\int p(x) d x} \\ \mu(x) & =C_2 e^{\int p(x) d x} . \end{aligned}

However, in practice when using these equations, why do we omit $C_2$? why does this not seem to affect our results? In essense, why are we only defining $u(x)$ as $\mu(x)=e^{\int p(x) d x}$ when performing our calculations?

Answer 1

when you're solving first-order linear ODE, or every ODE (or even PDE) that can be solved by using integration factor, you're looking for a $\mu$ that multiplying the differential equation by this factor make it possible for integration! so you actually multiply both side of equation by $\mu$. it means every non-zero constant factor of $\mu$ like $\mu'=c\mu$ would also works for equation, since it would be cancel out later from both side.

Answer 2

The constant is implicit in the indefinite integral notation, so need not be written explicitly. Remember $\int f(x)dx$ signifies all the antiderivatives of $f$ (each one only differing by a constant).