MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations
Problem
On a slope field for dxdy=f(x,y), every segment along the horizontal line y=2 is horizontal. Above that line the segments slope downward to the right, while below the line they slope upward to the right. Which statement is most strongly supported by these observations?
(a) y=2 is an equilibrium solution because dxdy=0 there, but nearby solutions move away from this line.
(b) y=2 is a semistable equilibrium: solutions approach from one side but move away on the other because the slopes have opposite signs above and below the line.
(c) y=2 is an equilibrium solution because dxdy=0 there, and nearby solutions approach this line from both sides.
(d) It is impossible to determine whether y=2 is a solution without an explicit formula for f(x,y).