Does it matter for 2nd order ODEs for b^2 - 4ac >0, which way the constants are?

i.e. getting a and b different to what mark schemes say etc.

I am fairly sure it doesn't make a difference, just want reassurance.

Also what is meant by the term "arbitrary constants" is that simply constants that do not change?

Do you mean A and B?
You need to have a, b and c the same(at least they have to have the same ratios).
Your A and B can be different to what the mark-scheme says. One example is if you use the opposite symbols to the mark-scheme(ie your A is their B and vice versa). This is fine because they're just symbols, you can use $\Gamma$ if you want. Another example is if you get a slightly different (correct) particular solution to them.
Arbitrary constants are just constants that can take any value but are fixed for any set of initial conditions(they're fixed at some arbitrary value).

As in m1 and m2, so what the exponential is being raised to. For general solutions I am sure that A and B are always fixed.

So arbitrary constants will exist for 2nd order DE's but not for 1st order DE's?

Thank you for your reply

That makes more sense.

We need to be clearer on what we mean by all of these symbols. In line with convention, I use a, b and c as the coefficients of the equation($a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)$). I use A and B as the arbitrary constants in the complementary function, $m_1, m_2$ as the roots of the auxiliary equation, so for the case you've given(b^2-4ac>0) the complementary function is $Ae^{m_1 x} + Be^{m_2x}$.
A and B are not fixed for the general solution. They are only fixed for a particular solution. a, b, c, $m_1, m_2$ are fixed in the general solution however.

Arbitrary constants exist for differential equations of any order. 1st order ones will have 1(A or c usually) and 2nd order ones will have 2(A and B or $\phi$ and A usually) etc..

quite often α & ß are used as the roots of the Auxiliary Equation.