An initial value problem fixes the first two coefficients: the value at the starting point is the constant term and the slope is the coefficient of the first power. Every other coefficient then follows from the recurrence, so the solution is determined term by term. When the starting point is not zero, the series is written in powers of the distance from it, and the same steps apply after a change of variable.

Dead reckoning at sea: from a known position and heading, each short leg follows from the one before, and the course unfolds from the start. The further from the start, the more legs are needed to stay on course.

$y''+xy=0$ with $y(0)=0$, $y'(0)=1$: $y=x-\tfrac{x^{4}}{12}+\tfrac{x^{7}}{504}-\cdots$. Near the start two terms are already close; further out each partial sum breaks away, and more terms are needed to follow the solution.

For $y''+p(x)y'+q(x)y=0$ with and given, write . Then and , and the recurrence gives every later coefficient. To use the method of the last lessons unchanged, substitute : the equation in is expanded about , with and rewritten in .

• $y''+xy=0$, $y(0)=0$, $$: , , and the recurrence gives .

The coefficient of the square is half the second derivative at the starting point. The equation, evaluated there, gives that second derivative from the value and the slope. Those are the first two coefficients, so the third follows from them without writing any series at all.

The coefficients are derivatives divided by factorials.

Evaluate the equation at the starting point.

Replace the value and the slope by the first two coefficients.

The value at the starting point is the constant term, and the slope is the coefficient of the first power.

Great work! You now know how to:

• fix the first coefficients from initial values
• expand about a starting point other than zero
• derive the third coefficient straight from the equation
• judge how far a partial sum can be trusted