Introduction to ODEs & Direction Fields
Differential Equations · Section 1.1 · generated practice set
Read an autonomous equation without solving it: equilibria, a sign chart, stability, and slopes straight off the direction field.
Practice this set → Fresh numbers on every attempt. No account needed.
Method
Section 1.1 is about reading an ODE without solving it. For an autonomous equation $y' = f(y)$, the whole long-term story is in the sign of $f$.
The three-step read
- Find equilibria. Set $f(y) = 0$ and solve. These are the constant solutions.
- Sign chart. On each interval between equilibria, pick a test value and record whether $f > 0$ (solutions rise) or $f < 0$ (solutions fall).
- Classify. Arrows pointing in from both sides → asymptotically stable. Out from both sides → unstable. In on one side, out on the other → semistable.
The derivative test is the shortcut: $f'(y^*) < 0$ means stable, $f'(y^*) > 0$ means unstable, $f'(y^*) = 0$ means go back to the sign chart.
For a direction field, remember the slope at a point $(t,y)$ is just $f(t,y)$ — you evaluate, you do not integrate.
Definitions and theorems
Worked example
Find and classify the equilibria of $y' = y^2 - 4y$.
- Set $y^2 - 4y = y(y-4) = 0$, so the equilibria are $y = 0$ and $y = 4$.
- $f'(y) = 2y - 4$.
- $f'(0) = -4 < 0$, so $y = 0$ is asymptotically stable.
- $f'(4) = 4 > 0$, so $y = 4$ is unstable.
- Sign check: on $(0,4)$ pick $y = 1$: $f(1) = -3 < 0$, solutions fall toward 0. Above 4, pick $y = 5$: $f(5) = 5 > 0$, solutions rise away. Consistent.
Common mistakes
- Solving instead of reading. You do not need the explicit solution to classify equilibria. Reaching for separation of variables here wastes half the exam period.
- Sign of $f'$, not sign of $f$. Stability is about how $f$ changes at the equilibrium, not its value there (which is always 0).
- Forgetting the equilibrium is a solution. "As $t\to\infty$, $y \to k$" is wrong if the initial value already equals $k$ — then $y \equiv k$ exactly.
- Semistable exists. A double root like $y' = (y-3)^2$ gives arrows in one side, out the other. Do not force every equilibrium into stable/unstable.
Practice it
Reading the method is not the same as being able to run it under time pressure. This set generates a new problem with new coefficients every attempt, marks each part separately, and shows the full worked solution.