The differential equation satisfied by the angular position of an ideal, frictionless pendulum is

\frac{d^{2}θ}{dt^{2}} + sin θ = 0.  (1)

If the pendulum is undergoing small oscillations around its equilibrium position (straight down, θ = 0), then θ is always small, and we can replace sin θ by the first term in its Taylor expansion:

\frac{d^{2}θ}{dt^{2}} + θ = 0.  (2)

The approximate equation (2) is linear, and we know how to write down the solution!

For a better approximation, we could keep the second nonvanishing term in the Taylor expansion:

\frac{d^{2}θ}{dt^{2}} + θ − \frac{1}{6} θ^{3} = 0.  (3)

Although (3) is hard to solve exactly, it is slightly easier to deal with than (1) (for example, in constructing an approximate solution by numerical methods or by perturbation theory).