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I have to linearize the following differential equation for small $\theta$ and $\dot \theta$. $$ a (\cos \theta) \ddot \theta +b (\tan \theta )\dot \theta +c \sin 3t \tan \theta+d\cos 3t\cos \theta+e\sin\theta=f\cos \theta,\tag{1} $$ where $a,b,c,d,e,f$ are constants.

My attempt:

Since $\theta$ is small so \begin{align*} \cos \theta & =1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots \approx 1\\ \sin \theta & =\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots \approx\theta \\ \tan \theta & = \theta+\frac{\theta^3}{3}+\frac{2\theta^5}{15}+\cdots \approx \theta. \end{align*}

So, the equation (1) reduces to $$ a\ddot \theta +b\theta \dot \theta+c(\sin 3t) \theta+d \cos t+e=f\cos 3t $$

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    Not positive, but I think you should drop the $b\theta\dot\theta$ term. (Otherwise your final equation is not linear) – 2017-01-13
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    Then how will I drop the term $b\theta \dot \theta $? – 2017-01-13
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    You can probably drop it since it is a product of two small numbers, but $b$ can't be large. – 2017-01-13
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    okay, yeah $b$ should not be large. thanks – 2017-01-13

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