# Difference between constants, arbitrary constants and variables in differential equation

The general solution of the differential equation $y''+\omega^2y=0$ can be written as: $$y=\alpha\cos{(\omega(t-c))}+\beta\sin{(\omega(t-c))}$$ Is it correct to say that:

1. $\omega$ and $c$ are constants
2. $\alpha$ and $\beta$ are arbitrary constants
3. $t$ is a variable

QUESTION: What is the difference between constants, arbitrary constants and variables?

• Yes, all three are correct. Apr 23, 2016 at 9:34
• You've got solution of DE. That's function depend of variable. Now exactly you've got family of solutions, because of arbitary constants (if there are initiall conditions, you could find this arbitary constants). So it's easy to say: variable - smth that you could change, arbitary constants - smth, that you could find (if there are initall conditions) and constants - some numbers from DE. Apr 23, 2016 at 9:42
• I don't see why $c$ is not arbitrary in the same sense that $\alpha$ and $\beta$ are. Usually, the solution is given by, either $\alpha \cos (\omega (t-c))$ or $\alpha \cos \omega t + \beta \sin \omega t$. In any case, you have two arbitrary constants, because your equation is second order. Apr 23, 2016 at 10:09

• How about the comment of @user40085? Why shouldn't $c$ be also an arbitrary constant? Meaning, isn't $c$ part of the initial conditions? Apr 23, 2016 at 10:35