13. Intro to Differential Equations

Shown above is a slope field for which of the following differential equations?

Which of the following differential equations could produce a slope field where the slope at each point $(x,y)$ is given by $y^3-x$?

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

Sketch a slope field for the following differential equation through the nine points shown on the graph. 

$y^{\prime }=y-xy$

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

b. In what regions are solutions increasing? Decreasing?

y'(t) = (y−1)(1+y)

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).

y'(t) = (y−2)(y+1)

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

c. Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing?

y'(t) = cos y for |y| ≤ π

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

b. In what regions are solutions increasing? Decreasing?

y'(t) = y(y+3)(4-y)

46–48. Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields.

Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′(t)+km(t)=I, where m(t) is the mass of the drug in the blood at time t≥0, K is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. Let I=10mg/hr and k=0.05 hr^−1.

c. What is the equilibrium solution?

What is the equilibrium solution of the equation y'(t) = 3y − 9? Is it stable or unstable?

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.

a. Find the equilibrium solutions. 

y′(t) = y(y - 3)(y + 2)