AP Calculus AB and BC
General vs Particular Solution
The general solution of a differential equation is the whole family of functions that satisfy it, written with an arbitrary constant. A particular solution is the single member of that family passing through a given initial condition.
General solution
Use when: No initial condition is given, or the question asks for the family of solutions.
Particular solution
Use when: An initial condition is given, such as a point the solution must pass through.
Side by side
| General | Particular | |
|---|---|---|
| Contains | An arbitrary constant | A specific constant |
| Describes | Infinitely many curves | One curve |
| Needs an initial condition | No | Yes |
| Matches a slope field | Every curve in it | One curve through the given point |
The workflow is fixed: separate and integrate to get the general solution, substitute the initial condition to solve for the constant, then write the particular solution with that value in place.
In a separable equation it is often cleaner to apply the initial condition right after integrating, before rearranging, because the constant is easier to isolate at that stage.
Solve for y
An answer left in implicit form is usually incomplete on the AP exam. Unless the question says otherwise, isolate the dependent variable before finishing.
Frequently asked questions
How many initial conditions do I need?
One for a first-order equation, which is all AP Calculus covers. Each integration introduces one constant that needs one condition to pin down.
In the CED: Unit 7: Differential Equations