Let's say I have a differential equation such as:
y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0
I can solve this (among other ways) by substitution and conversion to a system of differential equations:
w1 = y
w2 = y'
w1' = w2
w2' = 2tw2 - w1
w1(0) = 2.1
w2(0) = 1.0
Of course, if we start out with a system of differential equations, in certain cases, we can create a larger system via further substitutions.
My question is this: what if I start out with a system in which the highest order for the derivative of more than one variable appears in the same equation? For example:
q'' - 3r'q + r'' = 0
q'' + 5rq + 2r'' = 0
(I'm leaving out initial conditions here; just pick your own) I have been able to get it to work out for may problems by fudging things using Laplace transforms. But is there a substitution I can use on a problem like this? I've searched online and even some of the literature, but I can't seem to find anything. Maybe I'm just using the wrong terms. I realize there are other ways to do this. Just wondering if there's a substitution that will work.