27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.
c. Find the equilibrium points for the system.


x′(t) = −3x + xy, y′(t) = 2y − xy

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

c. Sketch the solution curve that corresponds to the initial condition y0=1. 

y′(t) = 2y + 4

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

c. Sketch the solution curve that corresponds to the initial condition y0=1. 

y′(t) = 6 - 2y

{Use of Tech} Free fall Using th e background given in Exercise 47, assume the resistance is given by f(v)=−Rv, for t≥0, where R>0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil).

c. Find the solution of this separable equation assuming v(0)=0 and 0<v<g/b.

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

c. Which time step results in the more accurate approximation? Explain your observations.

y′(t) = 4−y, y(0) = 3; y(t) = 4−e⁻ᵗ

Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.

c. Why is the condition A < T₀/2 needed?

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

d. Compare the errors in the approximations to y(T).

y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ