Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- A solution of a differential equation is a function. Usually there are infinitely many, forming a family.
- The slope field shows the whole family: every curve that follows the segments is one member.
- An initial condition (a point on the curve) selects one particular solution.
- To sketch it: start at the point, follow the segments right and left, keep the curve smooth and tangent to the segments, and stop at the edge of the field.
- A constant solution y = c is a horizontal line along which every segment is flat. Nearby solutions level off towards it or move away from it; they do not cross it.
- A curve of zero slopes that is not horizontal is not a solution.
Key relationships
| Question about a solution | Tool |
|---|---|
| Increasing or decreasing? | Sign of dy/dx |
| Relative max or min? | dy/dx = 0 with a sign change (or with d²y/dx² ≠ 0) |
| Concave up or down? | Sign of d²y/dx² (implicit differentiation, then substitute dy/dx) |
| Approximate value? | Tangent line at the known point; concave up → underestimate, concave down → overestimate |
| Constant solutions? | Values of y that make dy/dx = 0 for every x |
| Long run? | Which constant (or straight-line) solution the curves approach |
| dy/dx depends on x only | Solutions are vertical shifts: y = F(x) + C |
| dy/dx depends on y only | Solutions are horizontal shifts of each other |
Assumptions behind the method
- Sketches are estimates. Support every claim with the equation.
- For the equations in these pages, solution curves do not cross (a uniqueness result beyond the course). Justify in writing with slopes, not by quoting it.
Mistakes to avoid
- Sketching only to the right of the given point.
- Calling a sloping line of flat segments a solution.
- Drawing a curve that crosses a constant solution.
- Differentiating y as if it were a constant when finding d²y/dx².
- Leaving d²y/dx² in terms of dy/dx before evaluating.
- Saying a solution “reaches” a value it only approaches.
Quick self-check
- Find the constant solutions of dy/dx = y(y − 4). Is a solution with y(0) = 2 increasing or decreasing? (y = 0 and y = 4. At y = 2, dy/dx = 2(−2) = −4 < 0, so decreasing, and it falls towards y = 0.)
- For dy/dx = x + y, find d²y/dx² and its sign at (1, −3). (d²y/dx² = 1 + dy/dx. At (1, −3), dy/dx = −2, so d²y/dx² = −1: concave down.)
- Is y = 2x a solution of dy/dx = 2x − y? (No. Its slope is 2, but on that line the equation gives 2x − 2x = 0.)
Next: practice questions.