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15_pdfsam_math 54 differential equation solutions odd

15_pdfsam_math 54 differential equation solutions odd - the...

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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 defined for x < 0. 3. From Figure 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 field is shown in Figure B.4 in the answers section of the text. (b), (c) The direction field indicates that all solution curves (other than p ( t ) 0) will approach the horizontal line (asymptote) p = 1 . 5 as t + . Thus lim t + p ( t ) = 1 . 5 . (d) No. The direction field shows that populations greater than 1500 will steadily decrease, but can never reach 1500 or any smaller value, i.e., the solution curves cannot cross
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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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