Suppose that we have the following second-order ODE to solve: $$ay''+by'+cy = 0$$ The common way to finding the general solution is by finding the roots of the characteristic polynomial. Each root can be either real or complex. The question here is the following: If we were a scientist in 17th century and tried to solve that kind of problem, but we weren't aware of the "existence" of the complex numbers, how would we approach it? In other words, how can we find the general solution to that ODE by using methods of real analysis and avoiding the use of complex numbers?
Thanks for any answer in advance!