Derive the solution P(t) = MPo Ро + (М — Роје- of the logistic initial value problem P' = kP(M – P), -kMt P(0) = Po. Make it clear how your derivation depends on whether 0 < Po M.

Assume A population of tigers in a reserve grows according to the logistic differential equation. There are initially 30 tigers released but grown to a population of 50 after one year. It is estimated the reserve cannot support more than 600 tigers. Find an expression that will model the population of leopards at any time (t)

Suppose you want to find an equation for a curve that passes through the point (1, 2x3 y 2) and whose slope at any point (x, y) is A. B. C. You'd solve the differential equation Solving this differential equation with the initial condition, you get: E. 2y2 = x4 +7 نيا v2 = 2x4+2 ² = x4 +3 Op. ²x4+4 = D. 1²=14+3 p2 dx -2x³ = with initial condition y(1) = 2.

A firm uses capital (K) and labour (E) to produce output according to the production function Q(K,E)=lnK+lnE for K > 0 and E > 0, where ln is the natural logarithm. When either K = 0 or E = 0 or both, output is zero. The firm's cost function is C(K,E) = 9K^2+E^2. The firm wants to choose K and E so as to maximize its output subject to spending exactly $1,800. Solve the firm's constrained maximization problem and its Lagrangian function. Derive the first order conditions and find the candidate solution.The unique stationary point of the firm's constrained profit maximization problem is given by K*=? E*=?