Series Solutions

When a differential equation cannot be solved by elementary analytical methods (especially when it involves variable coefficients like x2y′′+y=0x^2 y'' + y = 0), we often assume the solution can be represented as an infinite power series.

Power Series Solution

We assume a solution of the form:
y(x)=∑n=0∞cn(x−x0)ny(x) = \sum_{n=0}^{\infty} c_n (x-x_0)^n
Typically, we expand the series around x0=0x_0 = 0 (a Maclaurin series):
y(x)=∑n=0∞cnxn=c0+c1x+c2x2+c3x3+…y(x) = \sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \dots

Convergence and the Ratio Test

A power series is only useful if it converges to a finite value. The interval of xx values for which the series converges is called the interval of convergence, centered at x0x_0, and half its length is the radius of convergence, RR.

Finding the Radius of Convergence

The primary tool to find RR is the Ratio Test. A power series ∑an\sum a_n converges absolutely if the limit of the absolute ratio of successive terms is less than 1:
lim⁡n→∞∣an+1an∣=L<1\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L < 1
Applying this to the power series an=cn(x−x0)na_n = c_n(x-x_0)^n:
  1. Set up the limit: lim⁡n→∞∣cn+1(x−x0)n+1cn(x−x0)n∣=lim⁡n→∞∣cn+1cn∣∣x−x0∣<1\lim_{n \to \infty} \left| \frac{c_{n+1}(x-x_0)^{n+1}}{c_n(x-x_0)^n} \right| = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| |x-x_0| < 1
  2. Solve for ∣x−x0∣|x-x_0|: ∣x−x0∣<lim⁡n→∞∣cncn+1∣=R|x-x_0| < \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right| = R
If R=0R=0, it converges only at x=x0x=x_0. If R=∞R=\infty, it converges for all real xx.

Power Series Convergence

Before diving into solving DEs with series, it helps to visualize how a power series converges to a target function as more terms are added within its radius of convergence. This is the foundation of Taylor series approximations.

Power Series Convergence Visualization

sin⁡(x)≈∑n=00(−1)nx2n+1(2n+1)!\sin(x) \approx \sum_{n=0}^{0} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
True Function
Approximation (1 term)
1 TermTerms: 115 Terms

Ordinary vs. Singular Points

Before solving a second-order linear DE, P(x)y′′+Q(x)y′+R(x)y=0P(x)y'' + Q(x)y' + R(x)y = 0, we must check if the expansion point x0x_0 is ordinary or singular.

Classification of Points

First, rewrite the DE in standard form: y′′+p(x)y′+q(x)y=0y'' + p(x)y' + q(x)y = 0, where p(x)=Q(x)/P(x)p(x) = Q(x)/P(x) and q(x)=R(x)/P(x)q(x) = R(x)/P(x).
  • Ordinary Point: If both p(x)p(x) and q(x)q(x) are analytic (have convergent Taylor series) at x0x_0. Practically, this means P(x0)≠0P(x_0) \neq 0. Use the standard power series method.
  • Singular Point: If P(x0)=0P(x_0) = 0 (meaning p(x)p(x) or q(x)q(x) is undefined at x0x_0).
    • Regular Singular Point: If the singularities are "mild." Specifically, if (x−x0)p(x)(x-x_0)p(x) and (x−x0)2q(x)(x-x_0)^2q(x) are both analytic at x0x_0. Use the Frobenius Method.
    • Irregular Singular Point: If the conditions for a regular singular point fail. Solutions behave wildly, and these methods generally do not apply.

Method of Power Series (Around an Ordinary Point)

  1. Assume Solution: Write y=∑n=0∞cnxny = \sum_{n=0}^\infty c_n x^n.
  2. Differentiate: Find y′=∑n=1∞ncnxn−1y' = \sum_{n=1}^\infty n c_n x^{n-1} and y′′=∑n=2∞n(n−1)cnxn−2y'' = \sum_{n=2}^\infty n(n-1) c_n x^{n-2}.
  3. Substitute: Plug y,y′,y′′y, y', y'' into the DE.
  4. Shift Indices: Adjust the summation indices (n→kn \to k) so all terms involve the exact same power of xx (e.g., xkx^k) and start at the same lower limit.
  5. Find Recurrence Relation: Combine the sums. By the identity property of series, the coefficient of every power xkx^k must equal zero. This gives an algebraic equation for ck+2c_{k+2} in terms of ckc_k and ck−1c_{k-1}, etc.
  6. Solve: Use the recurrence relation to find c2,c3,c4...c_2, c_3, c_4... in terms of arbitrary constants c0c_0 and c1c_1 (which correspond to initial conditions).

Method of Frobenius (Around a Regular Singular Point)

If x0=0x_0 = 0 is a regular singular point, standard power series fail. Instead, the solution takes the form of a Frobenius series, which adds an arbitrary power rr:
y(x)=∑n=0∞cnxn+r=xr∑n=0∞cnxny(x) = \sum_{n=0}^{\infty} c_n x^{n+r} = x^r \sum_{n=0}^{\infty} c_n x^n
where c0≠0c_0 \neq 0 and rr is a constant to be determined (the indicial root).
  1. Assume Solution: Write y=∑n=0∞cnxn+ry = \sum_{n=0}^{\infty} c_n x^{n+r}.
  2. Find Derivatives: Differentiate twice: y′=∑n=0∞(n+r)cnxn+r−1y' = \sum_{n=0}^{\infty} (n+r) c_n x^{n+r-1} and y′′=∑n=0∞(n+r)(n+r−1)cnxn+r−2y'' = \sum_{n=0}^{\infty} (n+r)(n+r-1) c_n x^{n+r-2}.
  3. Substitute into DE: Substitute into the differential equation, distribute any xx coefficients, and shift indices to align the powers of xx.
  4. Indicial Equation: Look at the term with the lowest power of xx (usually xr−2x^{r-2} or xrx^r) and set its coefficient to zero. Since we assumed c0≠0c_0 \neq 0, this yields a quadratic equation in rr, known as the indicial equation.
  5. Solve for r: Find the roots r1,r2r_1, r_2 (indicial roots). The form of the second linearly independent solution depends heavily on whether r1−r2r_1 - r_2 is an integer or zero.
Key Takeaways
  • Power Series Method: Assume y=∑cnxny = \sum c_n x^n to solve variable-coefficient DEs around an Ordinary Point (P(x0)≠0P(x_0) \neq 0).
  • Radius of Convergence (RR): Use the Ratio test to determine the interval ∣x−x0∣<R|x - x_0| < R where the infinite series approximation is mathematically valid.
  • Recurrence Relation: The core result of the method. It defines how to calculate higher-order coefficients cnc_n based on previous ones, usually leaving c0c_0 and c1c_1 as arbitrary constants.
  • Frobenius Method: Required when expanding around a Regular Singular Point (P(x0)=0P(x_0) = 0, but singularities are mild). Assumes y=xr∑cnxny = x^r \sum c_n x^n and requires solving an indicial equation for rr.
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