2. Given the system of differential equations a' = 0.2x -0.005xy, y'= -0.5y +0.01.xy, which models the rates of changes of two interacting species populations, describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction (competition, cooperation, or predation). Then find and characterize the system's critical points (type and stability). Determine what nonzero x- and y-populations can coexist. Then construct a phase plane portrait that enables you to describe the long term behavior of the two populations. Use https://www.geogebra.org/m/utcMvuUy to confirm your results.

1) For the DDS given by  Yn = 1/5 ( Yn-1)^2 + 6/5. Find the equilibrium values. Write your answers as a list, separated by commas. 1a) Ye = ?

Solve the system of the linear differential equations! メニムメと x'= 4X+y 4= 2x+ 3y with initial condition x(0)=4, yo)= 1 XCt)= ? XCt)=D7

Use variation of parameters to solve the given nonhomogeneous system. x-(::)*-() 0 8 8. X + 6. X' = -9t e X(t)

Solve the system of differential equations 'x'= 52x - 24y y'= 112x - 52y x(0) = 0, y(0) = -1 x(t) = y(t) =

Recast the system of higher order differential equations u" = 2u-3v v"=3tu'v'- cos u as a first order system.

3 Find the most general real-valued solution to the linear system of differential equations ' x1(t) = Ci + c2 x2(t)

Let f(x, y) = xy² and r(t) = (¹/1²,t³). Calculate Vf. (Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form (*, *). ) Vf= Calculate r' (t). (Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form (*, *). ) r' (t) = Use the Chain Rule for Paths to evaluateƒ(r(t)) at t = 1.5. (Use decimal notation. Give your answer to three decimal places.) d dt - ƒ(r(1.5)) = = Use the Chain Rule for Paths to evaluate ƒ(r(t)) at t = -1.7. (Use decimal notation. Give your answer to three decimal places.) d dt d f(r(-1.7)) =

The general form of a linear, homogeneous, second-order equation with constantcoefficients is (d^2 y/dt^2) + p(dy/dt) + qy = 0. (a) Write the first-order system for this equation, and write this system in matrixform.(b) Show that if q does not = 0, then the origin is the only equilibrium point of the system.(c) Show that if q does not = 0, then the only solution of the second-order equation with y constant is y(t) = 0 for all t.