AP Calculus AB and BC
Slope Field vs Solution Curve
A slope field draws the slope the differential equation assigns at each point, so it shows every solution at once. A solution curve is one function satisfying the equation, picked out of that family by an initial condition. Sketching a solution curve means following the segments through the given point.
Slope field
Use when: You need the general behaviour of solutions, or have to match a printed field to one of several differential equations.
Solution curve
Use when: An initial condition is given and the question asks you to sketch or find the one solution passing through that point.
Side by side
| Slope field | Solution curve | |
|---|---|---|
| What it shows | The slope at every point of a grid | One function that satisfies the equation |
| Built from | Evaluating at each grid point | The general solution with the constant pinned down |
| Needs an initial condition | No | Yes, that is what selects it |
| How many there are | One field per equation | One curve per initial condition |
| Reading it | Each segment is a direction, not a piece of a curve | Runs tangent to every segment it passes through |
| Common trap | Joining the segments into a single curve | Sketching a curve that misses the given point |
The relationship is one field to many curves. The equation assigns a slope to every point of the plane, and drawing a short segment at each grid point gives the field. Any curve that stays tangent to those segments wherever it travels is a solution, and there is one through each point.
The field and the general solution carry the same information in two formats. The arbitrary constant is what slides you from one curve in the family to the next, and an initial condition fixes it to a single value. That is why a field with no initial condition can never single out one curve.
Sketching on the exam
Start at the given point, move outward in both directions, and keep the curve parallel to the segments it passes. Stay inside the printed field, do not lift your pencil, and do not cross an equilibrium solution such as a horizontal line where .
Frequently asked questions
Can two solution curves cross?
No, not for the equations on the exam. Two curves crossing at an angle would need the equation to assign two different slopes at the crossing point, which it never does, and for well-behaved right sides (continuous, with a continuous -derivative) even a tangential meeting is ruled out, so exactly one solution passes through each point.
Do I have to solve the equation to sketch a solution curve?
No. Sketching needs only the field and the initial point, which is why slope field questions appear with equations that are hard to solve.
How does an initial condition pick out one curve?
It names a point the solution must pass through. Substituting that point into the general solution determines , and that value selects the single curve through it.
In the CED: Unit 7: Differential Equations