p1spring10_page 7 - ( x ) = sin( x ) x . 5b) If you...

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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 nd ( x-1 y ) = x-1 sin( x ). 5c) y ( x ) = x (Si( x ) + c ) 6a) This is separable, and you nd 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 nd it). Look at the slope eld 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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