Power Series Solutions

Precise definition

A power-series solution assumes $y(x)=\sum_{n=0}^\infty a_n(x-x_0)^n$ and differentiates term-by-term within its radius of convergence. Substitution into a linear ODE and equality of coefficients produce a recurrence for $a_n$. An ordinary point is one where the normalized coefficient functions are analytic.

Notation and mathematical language

$y'=\sum_{n=0}^\infty(n+1)a_{n+1}(x-x_0)^n$ and $y''=\sum_{n=0}^\infty(n+2)(n+1)a_{n+2}(x-x_0)^n$. Reindex all sums to the same power before equating coefficients. Initial values determine $a_0=y(x_0)$ and $a_1=y'(x_0)$.

Conceptual picture

The method replaces differentiation by algebra on an infinite coefficient sequence. Even and odd coefficients often decouple, revealing two independent solutions for a second-order equation. The nearest singularity of the coefficient functions often limits the convergence disk.

Conditions and key results

Termwise operations require convergence within a common radius, justified by power-series theory. Matching coefficients is valid when two power series agree on an interval. A truncated polynomial approximates the series solution; it is not the exact infinite solution and needs an error statement.

A reliable strategy

1. Choose the centre $x_0$ and verify it is an ordinary point for the method being used.
2. Write $y,y',y''$ with aligned powers and substitute them into the ODE.
3. Reindex sums, equate each coefficient to zero, and derive a recurrence.
4. Use initial data for free coefficients, list enough terms, and verify the recurrence or residual order.

Fully worked example

Interpretation and application

Series methods solve variable-coefficient equations without elementary closed forms and provide local approximations. Numerical use requires choosing a truncation and bounding or estimating the tail; symbolic equality refers to the infinite convergent series.

Common mistakes

Verification and reasonableness

• Substitute the truncated series and confirm the residual begins at the expected higher power.
• Check the recurrence for several consecutive indices and verify initial coefficients.
• Compare with a known solution or numerical solver inside the convergence interval.

Practice

1. What are $a_0,a_1$ when $y(0)=2,y'(0)=-1$?
2. For $y''+y=0$, what is the recurrence?
3. Is a five-term series exact?

Further deduction

At a regular singular point, an ordinary power series may fail, and the Frobenius form $y=\sum_{n=0}^\infty a_n(x-x_0)^{n+r}$ introduces an indicial exponent $r$. This is a distinct method with additional cases, especially when indicial roots differ by an integer. Identifying the point type prevents forcing the ordinary-point recurrence where its analyticity hypothesis is false.

The recurrence also proves parity. For $y''+y=0$, even coefficients depend only on $a_0$ and odd coefficients only on $a_1$. The two resulting series are linearly independent because their initial vectors $(y(0),y'(0))$ are $(1,0)$ and $(0,1)$. This connects coefficient algebra to the two-dimensional solution-space theorem.

Related topics

• Second-Order Linear Differential Equations
• Taylor Series
• Existence and Uniqueness