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Part II. Writing the equation d² ; =r_r dt2x in the form of the system v² d dt x = v, v=x-x² W(x): = x(t), x = x = = x(t), v = v(t), == (a) Find all the stationary points (x, v) (the points where d = 0, dt = 0). dt (b) Find the corresponding linear system near each critical point. (c) Find the corresponding linear system near each critical point. (d) Draw a phase portrait of the system near each critical point. (e) Draw a phase portrait taking into account the energy conservation, 1 +W(x) = const for each solution (x(t), v(t)) to system (2), 2 where the potential energy is given by the antiderivative of -x +x², x² x³ + 2 3 (1) (2)

Part II. Writing the equation d² ; =r_r dt2x in the form of the system v² d dt x = v, v=x-x² W(x): = x(t), x = x = = x(t), v = v(t), == (a) Find all the stationary points (x, v) (the points where d = 0, dt = 0). dt (b) Find the corresponding linear system near each critical point. (c) Find the corresponding linear system near each critical point. (d) Draw a phase portrait of the system near each critical point. (e) Draw a phase portrait taking into account the energy conservation, 1 +W(x) = const for each solution (x(t), v(t)) to system (2), 2 where the potential energy is given by the antiderivative of -x +x², x² x³ + 2 3 (1) (2)

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 (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), dx Find all the stationary points (x, v) (the points where dt = 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)
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), dx Find all the stationary points (x, v) (the points where dt = 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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