Question

please walk through this for me.

Let's modify logsic differantal equation of this example as followa 0.05P1- - ..natnn nf fish s . Tiithe r i Eh are caught (b) Draw a directio feld for this dfferential quation, Use the direction feld to sketch several splution ourves. 1200 1500 commersecaated list.) P eion initial ponuations F ,200, re For P-200. Pit)5elet sect For enn e'd For P S00, Pir) akct- w using partial fractions or nith a computer aloebra avatem, Use the initial pooulations 150 and 250. Selve this diffarential equation expiddly, either P-150 P 250 Graph the solutions nd compare with your skaches in part (b) 1000 1000 100 130 L00 120 s00 1000 100 20 20 co. 20

Answer 1

a) (-8) represents harvesting the fish at weekly rate. 8 fish/week is the rate at which fish are caught

b) Direction field is plotted below

1200 1000 800 600 400 200 0 0 10 20 40 50 60 70 80 90 30 100

Which corresponds to the last option

c) Equilibrium solutions are: (based on the above graph where the derivative is horizontal)

P(t)200,800

d) Graph of derivative function is:

10 f(x0.05.x| 1-1000 (800, 0) (200, 0) 200 800 0 400 600 1000 1200 10

And so

0 Po<200 P(t) is decreasing Po 200 P(t) is fixed 200 < Po800 P(t) is increasing Po 800 P(t) is fixed Po> 800 P(t) is decreasing

d) General solution to a) is $y(t)=200-\frac{600}{13\cdot\exp\left(-\frac{3t}{100}\right)-1}$

General solution to b) is $y(t)=200+\frac{600}{11\cdot\exp\left(-\frac{3t}{100}\right)+1}$

The two solutions are plotted below:

+ -500 X f(x)= 0.05x1l - 1000 -8 400 600 N g(x) 200- = -1 13 exp 100 300 X 600 Nh(x) =200+ 3x 11 exp 100 200 4 100 400 300 200 100 0 -100 200 powered by desmos 100

Which corresponds to option c) (the lower left option)

$\blacksquare$

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