A student says: 'I drew a direction field for y' = y and sketched one solution curve through (0, 1). But I could sketch a different curve through (0, 1) that doesn't follow the arrows as precisely — it would just be a slightly different solution.' What is wrong with this reasoning?
ADirection fields only show approximate behavior, so multiple curves through a point are possible
BUnder mild conditions, exactly one solution curve passes through each point; a second curve through (0, 1) would either cross the first or fail to be a solution
CThe direction field for y' = y doesn't apply at points where y ≠ 0
DDirection fields only apply to linear equations, not exponential ones
The existence and uniqueness theorem guarantees that under mild conditions on f, exactly one solution passes through each initial point. Any curve claiming to be a solution through (0, 1) must follow the direction field exactly. If a second 'solution' also passed through (0, 1), it would have to cross the first — but solution curves cannot cross, because crossing would require two different slopes at the same point, contradicting the differential equation.
Question 2 Multiple Choice
In an autonomous equation y' = f(y), all arrows in the direction field at height y = 3 are horizontal (slope = 0). What can you immediately conclude?
Ay = 3 is the only solution to the differential equation
By = 3 is an equilibrium solution — a constant function y(x) = 3 where f(3) = 0
CAll non-constant solutions eventually reach y = 3 and stay there
DThe direction field is undefined at height y = 3
Horizontal arrows mean the slope is zero at that height: f(3) = 0. This means y(x) = 3 is a constant solution (equilibrium) — if you start there, you stay there. But this does NOT mean all solutions are attracted to y = 3; stability depends on the signs of f(y) near y = 3, which you can read directly from the direction field.
Question 3 True / False
A direction field for y' = f(x, y) determines a single unique solution curve — the one that best fits most of the arrows in the field.
TTrue
FFalse
Answer: False
A direction field represents an entire family of solutions — one through each point in the plane. The field specifies the slope at every point, so any different initial condition gives a different solution curve. You need an initial condition to select a single solution from the family. The field shows the structure of all solutions simultaneously, which is its geometric power.
Question 4 True / False
Two distinct solution curves of the same differential equation y' = f(x, y) can never intersect, provided f satisfies the conditions of the existence and uniqueness theorem.
TTrue
FFalse
Answer: True
If two solution curves intersected at a point (x₀, y₀), both would pass through that point. But uniqueness guarantees exactly one solution through each point — so the two curves must be the same curve. Visually: solution curves can never cross, only flow alongside each other getting closer or farther apart.
Question 5 Short Answer
A direction field for an autonomous equation y' = f(y) shows arrows pointing upward above y = 2 and downward below y = 2, with horizontal arrows at y = 2. What does this tell you about solutions that start near but not at y = 2?
Think about your answer, then reveal below.
Model answer: y = 2 is an unstable equilibrium — solutions starting near but not at y = 2 are repelled away from it. Solutions above y = 2 increase (arrows point up) and solutions below y = 2 decrease (arrows point down), so both move away from the equilibrium.
Reading stability from a direction field is a key skill. Arrows pointing toward the equilibrium from both sides indicate stability (a sink); arrows pointing away indicate instability (a source). This geometric reading replaces algebraic analysis — you can classify equilibria just by observing the direction field, without solving the equation.