Differential Equations — Separation of Variables

A differential equation contains derivatives. You solve for the original function.

Method — Separation of Variables:

1. Get all y terms on one side, all x terms on the other

2. Integrate both sides

3. Solve for y (if possible)

4. Use initial condition to find C

Example: dy/dx = 2xy, y(0) = 3

dy/y = 2x dx

∫dy/y = ∫2x dx

ln|y| = x² + C

y = Aeˣ² where A = e^C

y(0) = 3: A = 3. So y = 3eˣ².

AP exam: Separation of variables appears on free response nearly every year. Show every step — the setup earns significant points.

Reference:

Wikipedia: Separation of Variables

https://en.wikipedia.org/wiki/Separation_of_variables