In this chapter we use a simple nonlinear ordinary differential equation as an example to introduce the basic ideas of the homotopy analysis method.

Consider a free sphere dropping in the air from a static state. Let t˜ denote the time, U(t˜) the velocity of the sphere, m the mass, and g the acceleration of gravity. Assume that the air resistance on the sphere is a U2(t˜), where a is a constant. Then, due to Newton’s second law, it holds

m dU(t˜) dt˜

= mg − aU2(t˜), (2.1)

subject to the initial condition

U(0) = 0. (2.2)

Physically speaking, the speed of a freely dropping sphere is increased due to the gravity until a steady velocity U∞ is reached. So, even not knowing the solution U(t˜) in detail, we can gain the limit velocity U∞ directly from (2.1), i.e.,

U∞ = √

mg

a . (2.3)

Using U∞ and U∞/g as the characteristic velocity and time, respectively, and writing

t˜ = (

U∞ g

) t, U(t˜) = U∞V (t), (2.4)

we have the dimensionless equation

V˙ (t) + V 2(t) = 1, t ≥ 0, (2.5) subject to the initial condition

V (0) = 0, (2.6)

denotes the with respect to t. Obviously, as t → +∞, i.e., t˜ → ∞ and U(t˜) → U∞, we have from (2.4) that

lim t→+∞V (t) = 1, (2.7)

even without solving Equations (2.5) and (2.6). The exact solution of Equations (2.5) and (2.6) is

V (t) = tanh(t), (2.8)

useful for the comparisons of different approximations.