AP Calculus AB and BC glossary

Separation of variables

Separation of variables solves a differential equation by moving all the y terms with dy to one side and all the x terms with dx to the other, then integrating both sides. It only works when the derivative factors into a function of x times a function of y.

dydx=g(x)h(y)  ⟹  ∫dyh(y)=∫g(x) dx\frac{dy}{dx} = g(x)h(y) \implies \int \frac{dy}{h(y)} = \int g(x)\,dx

You only need one constant of integration, conventionally written on the x side, because the difference of two constants is still just a constant.

The mistake

Treating dy over dx as a fraction without justification, or dividing by an expression that can be zero. Dividing by y loses the constant solution y equals zero.

Appears in: Unit 7: Differential Equations