1 d 0 1 t 1 bernoulli equation with n dt e let vp p1

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Unformatted text preview: /2 /2 1 t/2 dp 1 t/2 1 0 = (20+4p)) = 1 (4p p / t = 5. e 6. (a) Longterm behaviour: Slimvp(t) = 5.v p3Equilibrium solutions: 3p2(),)with IC p(0) = p0 . The DE is = (D e) = + e p (20+8 t 8 dt 4 42 4 Equilibriumdp solutions: 4p p3/21= t/2 p(4 p1/2 )2) p = 0, 16. 1 d 0) 1 t/ 1 Bernoulli equation with n ) =dt [e Let ]vp= p1 82e 2 (20+42 )) =0 = p1 p p3/22p0 . with IC p(0) = p0 . 6. (a) The DE is = (D S = 3/2. v = 3/ 3/ = p 1/ p ) v (4 2 3/ ), Using the (20+8 dt 4 4 4 DE: Equilibrium solutions: 4p p3/2et/2 v ) p(4 et/2 1/2 C) p = 0, 16. = 0 = 1 p+ ) 1 0 4 Bernoulli equation with n v= 3/2. 1 p 1/2 + 1p1 3/2 = p 1/2 ) v 0 = 1 p 3/2 p0 . Using the Let v = = 2 2 18 1 + Ce t/2 DE: 1 = 1 + C1 e t/2 = vv + = 4 1 8 4/2 Linear DE 1 0 v =2 p1 + 16 2 8 Standard form: v 0 p 1 v v 8 factor: General solution:+ 2= = 12. =Integrating t12 2 1e 1/2 dt = et/2 . /) (1 + Ce v + Linear DE = 1 t/2 16 1 t/2 8 4 2e t/2 0 ) 1. Apply IC: p(0) = p0 e) v 0+ 2 e v = 2 8 C = p p= p0 1/2 dt 1 1 (1 + C ) 0 Standard form: v + 2 v = d . t/2 Integrating 2factor: e = et/2 . 1 t/ 8 [e v ] 16 = e Solution of IVP: p(t) = dt 8 2. e (1 + ( 14 t/ 1)1 t/1)2 p v eC et/2 v 0 + t/2e0 2= =et/2 +t/2 e2 v 1 48 (b) Note that limt p(t) = 16. d t/2 1 t/2 t/2 1 If the initial price is greater than $16 the pricet/will 1 + Ce with limiting value $16. [ev v ] = + Cee 2 = decrease = 1 dt 48 If the initial price is equal to $16, it will remain at $16. 4 1 t/2 16 v price ewill+ C1 If the initial price is less than $16 2the = increase with limiting value $16. et/ General solution: p = v 2 = 4 t/2 )2 (1 + Ce 14 1 + Ce t/2 16 v C = p+ C11. t/2 = e )= Apply IC: p(0) = p0 ) p0 = (1 + C )2 4 4 4 p0 16 16 Solution of IVP: p(t) = . General solution: p = (1 + = 4 0 1)e t/t2/)2 2 v 2( p (1 + Ce 2 ) (b) Note that limt p(t) = 16. 4 16 b) (Discuss the(0) =is 0 ) p0 of your findings for p application. That is, $16. implications = ) C = decrease .with limiting value 1 Applyinitial price pgreater than $16 the price will the If the IC: p 2 p0 describe the price is equal to $16, it will remainand how this depends on p0 . behaviour of the(1 + Cin time at $16. price ) If the initial You need of IVP: is(less than $16 ) 16 If the initial price p t) = the increase with limiting value $16. Solution only consider p0 > 0. 4 price will 2 . (1 + ( p0 1)e t/2 ) Solution: (b) Note that limt p(t) = 16. 4 If the initial price is greater than $16 the price will decrease with limiting value $16. If the initial price is equal to $16, it will remain at $16. If the initial price is less than $16 the price will increase with limiting value $16. 4...
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