308_pdfsam_math 54 differential equation solutions odd

308_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 A sketch of this curve for a =0 . 003 and b =0 . 5 is shown in Figure B.39 in the answers of the text. (b) From the sketch in Figure B.39 in the answers of the text we see that the peak number of infected people is 295. (c) The peak number of infected people occurs when dI/dS = 0. From equation (5.27) we have dI dS =0= 1+ b a 1 S . Solving for S we obtain S = b a = 0 . 5 0 . 003 167 people . 35. (a) We denote v ( t )= x 0 ( t ) to transform the equation d 2 x dt 2 = x + 1 λ x to an equivalent system of two ±rst order di²erential equations, that is dx dt = v, dv dt = x + 1 λ x . (b) The phase plane equation in
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Unformatted text preview: xv-plane for the system in (a) is dv dx = − x + 1 / ( λ − x ) v . This equation is separable. Separating variables and integrating, we obtain v dv = ± − x + 1 λ − x ² dx ⇒ Z v dv = Z ± − x + 1 λ − x ² dx ⇒ 1 2 v 2 = − 1 2 x 2 − ln | λ − x | + C 1 ⇒ v 2 = C − x 2 − 2 ln | λ − x | ⇒ v = ± p C − x 2 − 2 ln( λ − x ) . (The absolute value sign is not necessary because x < λ .) 304...
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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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