# Advanced numerical solution of differential equation

• Apr 9th 2014, 07:03 PM
grandy
Advanced numerical solution of differential equation
Show that the explicit Runge-Kutta scheme
\frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})]

where $k_{1} = f(t,y_{n})$

applied to the equation $y'= y(1-y)$ has two spurious fixed points if $h>2$.

Briefy describe how you would investigate their stability.

=> my attempt so far
from $y'= y(1-y)$

$y'= 0$

$y=0$ or
$y=1$ which are the true fixed points.
after that i rearranged the runge kutta scheme
\frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})]

y_{n+1} = y_{n} + \frac{h}{2} [f(t,y_{n} + f(t+h, y_{n}+hf(t,y_{n})]

i try to put the fixed points into above scheme and try to get two two Spurious fixed point for $y_{n}$ but i got struck. for the stability to describe i need to get two Spurious fixed point first. but in general please help to describe stability too because this part i really get confuse. Anyone please help me, it will be really helpful for my other problems too if i got this answer correctly.