How to linearise a square root expression?
I have a non-linear ODE that I need to linearise.
I could go about linearising the trig terms using Taylor series...but not sure about a square root of the form:
$$\sqrt{(A+C)^2 + B -2 * \cos (\alpha - \theta)}$$
Thanks.
$\endgroup$ 32 Answers
$\begingroup$Since you have an ODE, you probably want to be linearizing around an equilbrium point $\theta_0$ (which might not be zero). Note that I'm assuming that $\theta$ is the variable.
Then think about the general form of a Taylor series expansion -- the linearization is given by $$ f(\theta) \approx f(\theta_0) + (\theta-\theta_0) f'(\theta_0) $$ where $$ f(\theta) = \sqrt{(A+C)^2 + B - 2\cos(\alpha - \theta)} $$
$\endgroup$ $\begingroup$If $(A+C)^2+B$ is large compared with 1 you can divide it out to get $$\sqrt{(A+C)^2+B}\sqrt{1-\frac{2cos(\alpha-\theta)}{(A+C)^2+B}}$$ and use $\sqrt{1-x}\approx 1-\frac{x}{2}$ for $x<<1$. If you are specifically interested in $\theta$ you can expand the cosine term and still use the approximation to the square root considering $\alpha$ to be fixed. Does that help?
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