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10Ma2aAnHw2 - 6[Problem 23.4 Let x be a fixed point of the...

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Ma 2a Fall 2010 (analytical track) PROBLEM SET 2 Due on Monday, October 11 I. Direction (slope) fields 1-2. Draw direction field diagrams for the following equations: (a) y 0 = y 2 - x , and (b) y 0 = y/ ( x 2 + y 2 ) . (You can use a computer.) II. Approximation of solutions 3. Consider the IVP ˙ x = 2 x , x (0) = 1. Approximate the value of x (1) by applying (i) Picard’s method of successive approximations n times, (ii) Euler’s line method with step h = 1 /n . Compute the limits as n → ∞ and find the value of x (1). 4. [See Example 21.2 in the text] Consider the IVP y 0 = y 2 - x , y (0) = 0. Approximate the value of y (2) by Euler’s method with step h = 1 / 2. (You can use calculators.) III. Difference equations 5. Draw a qualitatively correct cobweb diagram for the equation x n +1 = x 2 n + 1 4 . Describe the long term behavior of the orbits.
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Unformatted text preview: 6. [Problem 23.4] Let x * be a fixed point of the equation x n +1 = f ( x n ). Give an accurate definition of stability of x * . Prove that x * is stable if | f ( x * ) | < 1 and unstable if | f ( x * ) | > 1. Draw a cobweb diagram to illustrate stability in the case-1 < f ( x * ) < 0. IV. Separable and exact equations 7. Solve the initial value problem y = 3 x 2 3 y 2-4 , y (1) = 0 , and find the interval of the maximal solution. 8. [Problem 10.1] Check that the following equation is exact and hence solve it (find a first integral) (2 xy-sec 2 x ) dx + ( x 2 + 2 y ) dy = 0 . 1...
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