I am haveing trouble understanding the method of characteristics. I understand the derivation of the fact that the system $ \frac{dx}{dt}=a $,... gives us the characteristic curves and I understand what they represent.
My question is (geometrically): Why do we need to add an initial condition to these equations? i.e. - why isn't the system above enough ? (I think it is because we need the charactistic curves not to intersect each other, but I can't understand why the initla condition helps us in such a case... Can't we have the intersection in a point that is not on the initial curve?)
Thanks in advance