Question

A model that describes the population of a fishery in which harvesting takes place at a constant rate is given is dP/dt = kP - h where k and b are positive constants. (a) Solve the DE subject to P(0) = P₀.

Answer 1

The solution to the differential equation dP/dt = kP - h with initial condition P(0) = P₀ is an exponential function P = P₀ e^(kt - h t/C), where k and h are positive constants and C is the integration constant determined by P₀.

Step-by-step explanation:

The differential equation dP/dt = kP - h, where k and h are positive constants, represents the population of a fishery with constant harvesting. The solution to this differential equation subject to the initial condition P(0) = P₀ is derived using the method of separation of variables. By separating and integrating both sides, the solution can be expressed as an exponential function.

To solve the differential equation, we separate the variables by dividing both sides by P and then integrating:

1. Separate the variables: (1/P) dP = k dt - (h/P) dt
2. Integrate both sides: ∫ (1/P) dP = ∫ k dt - ∫ (h/P) dt
3. After integration, we obtain: ln|P| = kt - h t/C + C₁
4. Exponential both sides gives P(t): P = P₀ ekt - h t/C

Here, C₁ is the constant of integration determined by the initial condition P(0) = P₀. Thus, we arrive at a population model that accounts for growth and harvesting.