In numerical analysis, when considering a numerical method to solve a differential equation (think of Euler Forward, Euler Backward, Runge-Kutta 4, etc.), usually all the basic properties of the method (truncation error, stability, convergence) are derived in the case the differential equation to be solved is $y'=\lambda y$.
I understand that investigating this equation can be insightful, because it is an easy one to solve analytically. However, what I don't get, is that all the said properties (truncation error, stability, convergence) are seemingly understood to hold in the general case (i.e., any ODE, not necessarily $y'=\lambda y$). How do test function analysis results carry over to general ODE's?