4. One example.

Consider the equation form the last part


We expand y(x) in a perturbation series


Insertion in the equation yields


Insertion in the inital condition yields


Now let



that is

If we subtract 1 on both sides and divide the rest with  we get

On more time, we let


and if we repeat the procedure we realize that also

We determine y0, y1, y2,…. subsequently by solving the equations that we get by comparing terms containing the same power of  :


An approximate solution is thus