p1spring10_page 7

p1spring10_page 7 - x = sin x x 5b If you multiply by 1 x...

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some answers: 1b) When the dish is below 200C, its temperature ought to increase, so the derivative ought to be positive. Similarly if above 200C, the derivative ought to be negative. Both are corrected by putting a minus sign: T 0 = - 1 20 ( T - 200). 1c) T ( t ) = 200 - 100 e - t/ 20 2) equilibria: - 10, 0, 10. The 10 is seen to be stable by making a slope ﬁeld for example. Euler gives p 0 = 3 . 0, p 1 = 3 . 0273, p 2 = p 1 + (0 . 1)( p 1 / 10 - p 3 1 / 1000). 3b) y ( t ) = - 3 e - 2 t + 3 e - 3 t 3c) 1 2 ( y 0 ) 2 = - 3 2 + 1 2 y - 2 with y (0) = 1 2 4a) This cannot be in the form of f x dx + f y dy = 0 because you would have f xy = 0 6 = - 1 = f yx . 4b) It can be rearranged to dy dx = y , so the answer is 6 e x . 5a) by the Fundamental Theorem of Calculus, Si
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Unformatted text preview: ( x ) = sin( x ) x . 5b) If you multiply by 1 x , you ﬁnd ( x-1 y ) = x-1 sin( x ). 5c) y ( x ) = x (Si( x ) + c ) 6a) This is separable, and you ﬁnd y = 1 4 (2-x ) 2 as long as y > 0. 6b) The solution hits 0 when x = 2, so your formula becomes doubtful then (since you probably divided by y to ﬁnd it). Look at the slope ﬁeld or the DE itself to see that the formula in part a) is incorrect beyond x = 2, and that y = 0 is the correct way to continue the solution into that region. 7...
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This note was uploaded on 01/21/2011 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell.

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