Alternative method

An alternate (but mathematically equivalent) method is by repeated differentiation. This is illustrated by the following example.

Suppose we want to solve

As usual, we write

Now: if we just plug $x=0$ into the differential equation, we get

With the initial conditions, this means we have

From this, we already know

To get the next term in the Taylor series, we can differentiate the differential equation:

Plugging in $x=0$, we get

so that $y^{\prime \prime \prime }(0)=-y^{\prime \prime }(0)-y^{\prime }(0)=-3-2=-5$ and

and so on. Which method to use is a matter of preference, and also depends a bit on the form of the differential equation – this method is simpler when repeated differentiation is easy to perform.