Part II. Writing the equation d² [x = x − x², in the form of the system dx = = V, v=x-x² W(x) x = x(t), x= x(t), v = v(t), v² + W (x) = const for each solution (r(t), v(t)) to system (2), where the potential energy is given by the antiderivative of -x +x², x² = (a) Find all the stationary points (x, v) (the points where di = 0, d = 0). (b) Find the corresponding linear system near each critical point. (c) (d) Find the corresponding linear system near each critical point. Draw a phase portrait of the system near each critical point. Draw a phase portrait taking into account the energy conservation, (e) 2 + (1) co/8 (2)
Part II. Writing the equation d² [x = x − x², in the form of the system dx = = V, v=x-x² W(x) x = x(t), x= x(t), v = v(t), v² + W (x) = const for each solution (r(t), v(t)) to system (2), where the potential energy is given by the antiderivative of -x +x², x² = (a) Find all the stationary points (x, v) (the points where di = 0, d = 0). (b) Find the corresponding linear system near each critical point. (c) (d) Find the corresponding linear system near each critical point. Draw a phase portrait of the system near each critical point. Draw a phase portrait taking into account the energy conservation, (e) 2 + (1) co/8 (2)
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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Transcribed Image Text:Part II. Writing the equation
d²
dt²
in the form of the system
(a)
(b)
(c)
(d)
(e)
x= x= x²,
x = V₁
v = x= x²
dt
2
= x(t),
x =
x = x (t),
v²
+ W(x) = const for each solution (x(t), v(t)) to system (2),
v = v(t),
Find all the stationary points (x, v) (the points where d = 0, du = 0).
dt
Find the corresponding linear system near each critical point.
Find the corresponding linear system near each critical point.
Draw a phase portrait of the system near each critical point.
Draw a phase portrait taking into account the energy conservation,
1
where the potential energy is given by the antiderivative of -x + x²,
x²
W(x)
2
+
3
(1)
(2)
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