I'm just curious as to what the actual distinction means. I understand that the requirement for a linear ODE, is for all the coefficients to be functions of x (independent variable), and that all derivatives or y's (dependent variable) must be of degree one, but that doesn't tell me much. In a normal function, there is a clear distinction between a linear and a nonlinear one. For example, y = 3x, it's clear here that y changes linearly with x, and is always three times as big as x. On y = x^2, it's obvious that the change is not linear...so the relationship isn't linear. Now how can I analyze linear differential equations and nonlinear differential equations in a similar manner? How does their behavior, or their 'meaning' differ?