7.6 Separation of Variables

The method

For dy/dx = g(x)·h(y), separate the variables: move all y-terms with dy to one side and all x-terms with dx to the other — (1/h(y)) dy = g(x) dx.

Integrate both sides

Integrate both sides, add a + C, and solve for y. This gives the general solution (a family of curves depending on C). Example: dy/dx = xy separates to (1/y) dy = x dx.

Key terms

Separation of variables

Separate y-terms (dy) from x-terms (dx).

Applicable form

dy/dx = g(x)·h(y).

Steps

Separate, integrate both sides, add C, solve for y.

General solution

A family of solutions with constant C.

Example

dy/dx = xy → (1/y) dy = x dx.

Try an AP-style question

What is the general solution of dy/dx = xy?

1. y = Ce^x
2. y = Ce^(x²/2)
3. y = x²/2 + C
4. y = Cx