Bring the chapter together: substitute a series and shift its indices, find and unwind a recurrence, fix the first coefficients from initial values, locate the singular points and the guaranteed reach, and find the exponents at a regular singular point. No worked example sits above the answers.

An accountant balancing books line by line: every entry has its place, and the totals must agree in every column.

Decide first what is asked. A series solution: substitute $\sum a_{n}x^{n}$, shift so that every sum runs over $x^{n}$, set each coefficient to zero and solve for the highest index. Initial values give $a_{0}$ and $a_{1}$. Reach: at least to the nearest singular point, complex ones included. At a regular singular point: $x^{r}\sum a_{n}x^{n}$, with $r$ from the indicial equation.

• $y''\mapsto\sum(n+2)(n+1)a_{n+2}x^{n}$, $xy'\mapsto\sum na_{n}x^{n}$, $xy\mapsto\sum a_{n-1}x^{n}$.
• A recurrence from $a_{n}$ to $a_{n+2}$ splits the coefficients into an even and an odd family.
• A zero factor in the recurrence stops a family: a polynomial solution.
• Indicial equation: $r(r-1)+p_{0}r+q_{0}=0$, with and the values at of and .