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MIT18_03S10_5ex

# MIT18_03S10_5ex - Section 5 Graphing Systems 5A The Phase...

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± Section 5. Graphing Systems 5A. The Phase Plane 5A-1. Find the critical points of each of the following non-linear autonomous systems. x = x 2 y 2 x = 1 x + y a) b) y = x xy y = y + 2 x 2 5A-2. Write each of the following equations as an equivalent first-order system, and find the critical points. a) x �� + a ( x 2 1) x + x = 0 b) x �� x + 1 x 2 = 0 5A-3. In general, what can you say about the relation between the trajectories and the critical points of the system on the left below, and those of the two systems on the right? x = f ( x, y ) x = f ( x, y ) x = g ( x, y ) a) y = g ( x, y ) y = g ( x, y ) b) y = f ( x, y ) 5A-4. Consider the autonomous system x = f ( x, y ) x ( t ) ; solution : x = . y = g ( x, y ) y ( t ) a) Show that if x 1 ( t ) is a solution, then x 2 ( t ) = x 1 ( t t 0 ) is also a solution. What is the geometric relation between the two solutions? b) The existence and uniqueness theorem for the system says that if f and g are contin- uously differentiable everywhere, there is one and only one solution x ( t ) satisfying a given initial condition x ( t 0 ) = x 0 . Using this and part (a), show that two trajectories cannot intersect anywhere. (Note that if two trajectories intersect at a point ( a, b ), the corresponding solutions x ( t ) which trace them out may be at ( a, b ) at different times. Part (a) shows how to get around this diﬃculty.) 5B. Sketching Linear Systems x = x 5B-1. Follow the Notes (GS.2) for graphing the trajectories of the system y = 2 y . dy a) Eliminate t to get one ODE = F ( x, y ). Solve it and sketch the solution curves. dx b) Solve the original system (by inspection, or using eigenvalues and eigenvectors), obtaining the equations of the trajectories in parametric form: x = x ( t ) , y = y ( t ). Using these, put the direction of motion on your solution curves for part (a). What new trajectories are there which were not included in the curves found in part (a)?

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MIT18_03S10_5ex - Section 5 Graphing Systems 5A The Phase...

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