(d2x/dt2)+2(dx/dt)-3x+x3=0(a) Convert this second-differential equation into a first-order system in terms of x and v, where v = dx/dt.(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. dx1.dr?dx– 3x + x³ = 0dt(a) Convert this second-differential equation into a first-order system in terms of x and v, wherev = dx/dt.(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.

We are given : dx2dt2+2dxdt-3x+x3=0---(1) v=dxdt---(2)⇒dvdt=dx2dt2--(3) Using (2),(3) in (1), we…

Answered: d²x dr? dx + 2 - 3x + x = 0 dt (a)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(d²x/dt²)+2(dx/dt)-3x+x³=0

(a) Convert this second-differential equation into a first-order system in terms of x and v, where v = dx/dt.
(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.

dx
1.
dr?
dx
– 3x + x³ = 0
dt
(a) Convert this second-differential equation into a first-order system in terms of x and v, where
v = dx/dt.
(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 :

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

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

Using (2),(3) in (1), we have

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

Hence, the 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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