1. ## 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.

2. 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 $\lim_{x\to 0}\frac{sin x}{x}= 1$ and that $sin(x-\pi)= -sin(x)$ so $\lim_{x\to \pi}\frac{sin (x-\pi)}{x-\pi}= \lim_{x\to 0}\frac{-sin(x)}{x}= -1$ so that sin x is approximately $-(x-\pi)$ for x close to $\pi$.

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.