Answered: dx 3x + x = 0 dt (a) Convert this second-differential equation into a first-order system in terms of x and v, where v = dx/dt. %3D (b) Find all equilibrium…

Name: please help me this question! Thank you a lot X.dx3x + x' = 0+ 2-dt?1.
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Description: please help me this question! Thank you a lot X.dx3x + x' = 0+ 2-dt?1.dt(a) Convert this second-differential equation into a first-order system in terms of x and v, wheredx/dt.V =(b) Find all equilibrium points.(c) Use phase plane plotter (Bluffton)

Advanced Engineering Mathematics

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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please help me this question! Thank you a lot

X.
dx
3x + x' = 0
+ 2-
dt?
1.
dt
(a) Convert this second-differential equation into a first-order system in terms of x and v, where
dx/dt.
V =
(b) Find all equilibrium points.
(c) Use phase plane plotter (Bluffton) to sketch the associated direction field.
(d) Use the direction field to make a rough sketch of the phase portrait of the system.

Expert Solution

Part(a)

We are given :
$\frac{d {}^{2}x}{d t^{2}} + 2 \frac{d x}{d t} - 3 x + x^{3} = 0 - - - (1)$

Now, $v = \frac{d x}{d t} - - - (2)$

$\Rightarrow \frac{d v}{d t} = \frac{d {}^{2}x}{d t^{2}} - - - (3)$

Hence, using (2) and (3) , (1) can be written as,

$\Rightarrow \frac{d v}{d t} + 2 v - 3 x + x^{3} = 0$

Hence, the required first order system is:

$\frac{d v}{d t} + 2 v - 3 x + x^{3} = 0, w h e r e v = \frac{d x}{d t}$

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