Consider the equation y' = y(4-y)-3. This equation describes, e.g., growth of a population of fish...

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Consider the equation y' = y(4-y)-3. This equation describes, e.g., growth of a population of fish in a pond assuming that 3 units of fish is caught per unit of time. a) (1 pt) Sketch the direction field and/or the phase line. b) (1 pt) Find equilibrium states and determine their stability. c) (1 pt) For which initial values y(0) = yo the solution tends to the stable equilibrium point? d) (1 pt) Solve this equation using dsolve method in Python sympy library. e) (1 pt) Explain why the formula from the previous part does not describe all solutions. Modify the formula to cover more solutions and list all "exceptional" solutions that are not given by this formula. f) (1 pt) Use the formula from part 2e to solve the initial value problem for y(0) = 0.5. g) (1 pt) Note that the formula from part 2f tends to the stable equilibrium point as t→∞ while the answer to part 2c does not include 0.5. Explain why there is no contradiction here. Hint: plot the solution in Python or Desmos. Consider the equation y' = y(4-y)-3. This equation describes, e.g., growth of a population of fish in a pond assuming that 3 units of fish is caught per unit of time. a) (1 pt) Sketch the direction field and/or the phase line. b) (1 pt) Find equilibrium states and determine their stability. c) (1 pt) For which initial values y(0) = yo the solution tends to the stable equilibrium point? d) (1 pt) Solve this equation using dsolve method in Python sympy library. e) (1 pt) Explain why the formula from the previous part does not describe all solutions. Modify the formula to cover more solutions and list all "exceptional" solutions that are not given by this formula. f) (1 pt) Use the formula from part 2e to solve the initial value problem for y(0) = 0.5. g) (1 pt) Note that the formula from part 2f tends to the stable equilibrium point as t→∞ while the answer to part 2c does not include 0.5. Explain why there is no contradiction here. Hint: plot the solution in Python or Desmos.

Answer rating: 100% (QA)

a To sketch the direction field we need to plot a grid of points in the yt plane and then draw short line segments with slopes equal to y4y3 at each p... View the full answer