# direct linearisation

• Dec 6th 2009, 01:33 PM
mamt6
direct linearisation
Use the method of direct linearisation to determine the stability of
(a) the fixed point at x = pi for the differential equation dx/dt = sin x, and
(b) the only fixed point for the differential equation dx/dt = ln (x/a) , a > 0.
(Hint: ln(1 + z) ~ z for |z| < 1.)
In each case verify your answer using the method of linearisation by Taylor polynomial.
• Dec 10th 2009, 06:45 AM
HallsofIvy
Quote:

Originally Posted by mamt6
Use the method of direct linearisation to determine the stability of
(a) the fixed point at x = pi for the differential equation dx/dt = sin x,

Two things you learned long ago are that $\displaystyle \lim_{x\to 0}\frac{sin x}{x}= 1$ and that $\displaystyle sin(x-\pi)= -sin(x)$ so $\displaystyle \lim_{x\to \pi}\frac{sin (x-\pi)}{x-\pi}= \lim_{x\to 0}\frac{-sin(x)}{x}= -1$ so that sin x is approximately $\displaystyle -(x-\pi)$ for x close to $\displaystyle \pi$.

Quote:

and
(b) the only fixed point for the differential equation dx/dt = ln (x/a) , a > 0.
(Hint: ln(1 + z) ~ z for |z| < 1.)
In each case verify your answer using the method of linearisation by Taylor polynomial.
x/a= 1+ (x/a- 1) so ln(x/a) is approximately x/a- 1. That's the linearization you want.