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10Ma2aAnHw1

10Ma2aAnHw1 - Ma 2a(analytical track Fall 2010 PROBLEM SET...

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Ma 2a Fall 2010 (analytical track) PROBLEM SET 1 Due on Monday, October 4 [All references are to J. C. Robinson] I. 1D autonomous equations (formula solution) 1. [Problem 8.7] Show that for k 6 = 0 the solution of the IVP ˙ x = kx - x 2 , x (0) = x 0 , is x ( t ) = ke kt x 0 x 0 ( e kt - 1) + k . Using this explicit solution describe the behavior of x ( t ) as t → ∞ for k < 0 and k > 0. 2. [Problem 8.11] Assuming that f ( x ) is continuously differentiable, show that if the solution of ˙ x = f ( x ) , x (0) = x 0 blows up to x = + in finite time, then Z x 0 dx f ( x ) < . II. 1D autonomous equations (qualitative approach) 3-4. [Problem 7.1] For each of the following equations draw the phase diagram, labeling the stationary points as stable or unstable: (iii) ˙ x = (1 + x )(2 - x ) sin x , (v) ˙ x = x 2 - x 4 . 5. [Problem 7.9] Shaw that for autonomous scalar equations, if a stationary point is attracting, then it must also be stable.

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10Ma2aAnHw1 - Ma 2a(analytical track Fall 2010 PROBLEM SET...

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