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Writing the equation d² dt2 in the form of the system +x², = x³ X = dt V, d ·V= x³ + x4 dt x = x (t), x = x(t), v = v(t), dx (a) Find all the stationary points (x, v) (the points where d = 0, dt dv dt (1) (2) = 0).

Writing the equation d² dt2 in the form of the system +x², = x³ X = dt V, d ·V= x³ + x4 dt x = x (t), x = x(t), v = v(t), dx (a) Find all the stationary points (x, v) (the points where d = 0, dt dv dt (1) (2) = 0).

Advanced Engineering Mathematics
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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Part II. Writing the equation d² dt2 in the form of the system dx d dt (c) (d) (e) = V₂ ·V= = x³ + x², X = x³ + x² W(x): x = x (t), = x = x(t), v = v(t), (a) Find all the stationary points (x, v) (the points where d = 0, dt (b) 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, 12/201² v² + W(x) = const for each solution (x(t), v(t)) to system (2), where the potential energy is given by the antiderivative of −x³ – x¹, x4 10/2/20 dv dt x5 (1) (2) = 0).
Transcribed Image Text:Part II. Writing the equation d² dt2 in the form of the system dx d dt (c) (d) (e) = V₂ ·V= = x³ + x², X = x³ + x² W(x): x = x (t), = x = x(t), v = v(t), (a) Find all the stationary points (x, v) (the points where d = 0, dt (b) 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, 12/201² v² + W(x) = const for each solution (x(t), v(t)) to system (2), where the potential energy is given by the antiderivative of −x³ – x¹, x4 10/2/20 dv dt x5 (1) (2) = 0).
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