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3. Linearize the following non-linear dynamic models.
(a) 2 =ay + py? +ylny a, 3, y: constants
(b) 2 = om + ym?sinam a: constant
© =ym-—= 2y+m
(@ oa = 2yf + 3y^1y; + my — m^3
ic =y; — 4mymynr?
(e) 222 = 32 = yiy, — 0.5y,my + myyi
2% + & = Iny, + yjcos^2my — my,
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Step 1
a\_5' = 2a\_1mm' + a\_2m'' - a\_3m^2m'' + a\_4m^2\frac{m'''}{m'}
Linearize the given non-linear dynamic models. \end{align*} \end{align*} Therefore, the linearized model is \end{align*} \end{align*} Therefore, the linearized model is \end{align*} \end{align*} Therefore, the linearized model is \end{align*} \end{align*} Therefore, the linearized model is \end{align*} \end{align*} Therefore, the linearized model is
Final Answer
The linearized models for the given non-linear dynamic models are: (a) $a\_2' = a\_1 + a\_3 y'' + a\_4 y' + a\_4 ln(y) + a\_4 - \frac{a\_4}{y}$ (b) $a\_2' = a\_1 m' + a\_3 m''sin(a\_4) + a\_3 m'cos(a\_4)a\_4'$ (c) $a\_3' = a\_1 m' - a\_2 m'' - 2a\_2 mm'$ (d) $a\_4' = a\_1 f'(m)m' + a\_2 m'' + a\_3 m' - 3a\_5 m^2 m'$ (e) $a\_5' = 2a\_1 mm' + a\_2 m'' - a\_3 m^2 m'' + a\_4 m^2\frac{m'''}{m'}$
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