A predictor-corrector method and stability

[wel]

A predictor-corrector method for the approximate solution of y'=f(t,y) uses
\begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P
\end{equation}
as predictor and
\begin{equation} y_{n+1}-y_{n}=\frac{h}{2}(f_{n+1}-f_{n}) \tag C
\end{equation}
IF (P) and (C) are used in PECE mode on the vector problem
\begin{equation} \frac{du}{dt}=u
\end{equation}
\begin{equation} \frac{dv}{dt}=-10u-11v+cos(2\pi t)
\end{equation}
with u(0),v(0) given, find the largest constant \gamma >0 for which the scheme is stable in the sense of Von Neumann (Fourier series stability and frequency) whenever 0<\gamma<0. Give full details of your argument.

=>
I haven't try very well because its really difficult question for me.
I was thinking
\begin{equation} y_{n+1}=y_{n}+hf_{n} \tag P
\end{equation}
as predictor and
\begin{equation} y_{n+1}=y_{n}+\frac{h}{2}(f_{n+1}-f_{n}) \tag C
\end{equation}
iam trying to get transition matrix but these condition
\begin{equation} \frac{du}{dt}=u
\end{equation}
\begin{equation} \frac{dv}{dt}=-10u-11v+cos(2\pi t)
\end{equation}
i don't know how and where to use.
please help me.

 

[HallsofIvy]

Are you saying you do not know how to solve for u and v in those last two equations?
\frac{du}{dt}= u
should be easy. And once you have that, the second equation if
\frac{dv}{dt}+ 11v= -10u+ cos(2\pi t)
is a relatively easy "linear equation with constant coefficients".