# 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). – Dr. Lutz Lehmann Jul 10 '18 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.