# How do test function analysis results carry over to general equations

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?

• Equations that do not behave as nice as the test cases (if the step size is chosen appropriately) are called stiff (for the method and number type used). Commented Jul 10, 2018 at 16:39

Consider $y'(x) = f(y,x)$ with $y(x_0)=b$. Then consider: $$y'(x) \approx f(b,x)+\frac{\partial f}{\partial y}\bigg\rvert_{y=b} [y-b]\approx \lambda y(x) + f(b,x) - \lambda b = \lambda y + c$$ where we have linearized about $y=b$, fixed some $x=x_0$, and let $\lambda=\partial_y f(b,x_0)$. Then of course we can solve the homogeneous ODE $y' =\lambda y$ to get a general solution locally.