15_pdfsam_math 54 differential equation solutions odd

15_pdfsam_math 54 - the line p = 1 5 Indeed the constant Function p t 1 5 is a solution to the given logistic equation and the uniqueness part

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Exercises 1.3 (d) As x →∞ or x →−∞ , the solution in part (b) increases unboundedly and has the lines y =2 x and y = 2 x , respectively, as slant asymptotes. The solution in part (c) also increases without bound as x →∞ and approaches the line y =2 x , while it is not even defned For x< 0. 3. ±rom ±igure B.3 in the answers section oF the text, we conclude that, regardless oF the initial velocity, v (0), the corresponding solution curve v = v ( t ) has the line v = 8 as a horizontal asymptote, that is, lim t →∞ v ( t ) = 8. This explains the name “terminal velocity” For the value v =8 . 5. (a) The graph oF the directional feld is shown in ±igure B.4 in the answers section oF the text. (b), (c) The direction feld indicates that all solution curves (other than p ( t ) 0) will approach the horizontal line (asymptote) p =1 . 5as t + . Thus lim t + p ( t )=1 . 5. (d) No. The direction feld shows that populations greater than 1500 will steadily decrease,
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Unformatted text preview: the line p = 1 . 5 . Indeed, the constant Function p ( t ) ≡ 1 . 5 is a solution to the given logistic equation, and the uniqueness part oF Theorem 1, page 12, prevents intersections oF solution curves. 6. (a) The slope oF a solution to the di²erential equation dy/dx = x + sin y is given by dy/dx . ThereFore the slope at (1 , π/ 2) is equal to dy dx = 1 + sin π 2 = 2 . (b) The solution curve is increasing iF the slope oF the curve is greater than zero. ±rom part (a) we know the slope to be x + sin y . The Function sin y has values ranging From − 1 to 1; thereFore iF x is greater than 1 then the slope will always have a value greater than zero. This tells us that the solution curve is increasing. (c) The second derivative oF every solution can be determined by fnding the derivative oF 11...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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