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Woohoo, I'm in diff eq now. Gosh, I've been on this board since Calculus I think. Sorry I usually only come around when I have a question. :\

Sooooo, Thursday my professor did an example of solving a system of linear differential equations with using the linear operator D.

There was no explanation involved, just "Here's how you solve this one."

So, now I'm doing my homework, and I need to clarify, and this might seem like a silly question, but bear with me.

Is the whole idea just to get the system into a form that is solvable only by linear methods?

For example, if you have a system with 2 equations, in 2 unknowns, say x and y, and you eliminate y, you get it into a form where you will solve for x, then rinse, lather, repeat, go back eliminate and solve for y.

If after eliminating x or y, it can be any order, right? And, if it's first order, it will be a linear or Bernoulli form, if it's second order, it will be undetermined coefficients or variation of parameters form, etc.

It seems to me that since it's a linear system, the solution should ONLY be solvable by linear means. No exact, no separable, no homogeneous method. Just linear methods.

Is this correct?