Solution for non-omogeneus first-order linear "nil" coefficient differential equation I know that we can identify a specific family of the differential equations called:
Heterogeneous first-order linear constant coefficient ordinary differential equation
(Non-homogeneous = Heterogeneous ?)
The generic form is something like:
$\displaystyle y' + cy = f(x)$
I have a differential equation like this:
$\displaystyle {1 \over 2 \pi}{d\alpha(t) \over dt} = k_f m(t)$
So this is of the aforementioned form with $c = 0$.
I don't know if this can be called a non-homogeneous first-order linear "nil" coefficient differential equation.
Now I need to re-write/solve the equation to get the value of $\alpha(t)$.
The solution is the following:
$\displaystyle \fbox{$ \alpha(t) = 2 \pi k_f \int_{-\infty}^t m(\theta) d\theta $}$
I guess that this is the result of applying the integral to both sides of the equation but I can't figure out why the integral in the right part is like that. Why the bounds of the integral are $(-\infty, t]$ ? Is there a specific rule for this ?
 A: 
I don't know if this can be called a non-homogeneous first-order linear "nil" coefficient differential equation.

$0$ is a constant, so it's still an example of a non-homogeneous (or "heterogeneous" or "inhomogeneous") first-order linear constant-coefficient differential equation. In my dialect of English, "nil" for "zero" is very rare. And I wouldn't even be confident of the meaning of "linear zero coefficient differential equation" because I'd have to think through which coefficient(s) are likely to be zero.

Why the bounds of the integral are $(-\infty,t]$?

Certainly, a different lower bound (like $7$ instead of $-\infty$) would still yield a solution to the differential equation. So either the answer is "no reason, that's just one possible way to get an antiderivative" or the answer is "something about Physics convention or the physical intuition for the context makes it so that ${\displaystyle \lim_{t\to-\infty}} \alpha(t)=0$ is reasonable. I don't recognize this differential equation, so I don't know which of those two is the case.
