• 6.1 Solutions About Ordinary Points
  • 6.2 Solutions About Singular Points
  • 6.3 Two Special Equations
  • Series Solutions of Linear Equations About a Singular Point (Method of Froebenius)

Series Solutions of Linear Equations About an Ordinary Point

Procedure for finding a general solution, in terms of a Power Series, of a second-order linear DE, around the Ordinary Point $x0 = 0.$

  1. Put the DE in the standard form \[ y^{\prime\prime} + P\left(x\right) y^\prime + Q\left(x\right) y = 0. \]
  2. Substitute the series solution and its first two derivatives into the DE in 1. That is, substitute the terms \[ \eqalign{ &y=\sum_{n=0}^{\infty}{ \quad c_nx^n } \\ &y^\prime=\sum_{n=1}^{\infty}{ \quad n c_nx^{n-1} } \\ &y^{\prime\prime}=\sum_{n=2}^{\infty}{ \quad\left(n-1\right)c_nx^{n-2} } } \] into the equation to get: \[ \sum_{n=0}^{\infty}{ \quad\left(n-1\right) c_n x^{n-2} } + P\left(x\right) \sum_{n=0}^{\infty}{ \quad n c_nx^{n-1} } + Q\left(x\right) \sum_{n=0}^{\infty}{ \quad c_nx^n } = 0 \]
  3. Distribute the factors $P(x)$ and $Q(x)$ so they are inside the sigma, then combine like terms, in particular the $x$ terms.
  4. Get the sigmas "in phase" by making an appropriate substitution to cause the exponents of $x$ in each series to be equal.
  5. Make the starting index of each series equal by, for instance, evaluating the first few terms of those series with smaller starting indices, or, say, by lowering the index of a series if it does not change the series.
  6. Combine the sigmas into one and factor out the $x$ term.
  7. Use the identity property on p. 269 to derive the recurrence relation and other equations that give the coefficients of $x$ in terms of 0.
  8. Find as many of the first terms of the series as desired, and use the recurrence relation and other equations in 7 to express the coefficients of each term in terms of either $c0$ or $c1.$
  9. Factor out the coefficients c0 and c1 so the series are written in the form \[ y = c_0 y_1\left(x\right) + c_1 y_2\left(x\right) \] This is the general solution of the DE in 1.

Step 2 is justified by theorem 6.1 which says that the second-order linear DE \[ y^{\prime\prime}+P\left(x\right)y^\prime+Q\left(x\right)y=0 \] has at least two linearly independent solutions of the form \[ y = \sum_{n=0}^{\infty}{\quad c_nx^n}. \]

Regular and Irregular Singular Points of DE’s with Polynomial Coefficients

Singular Point

If $a_2\left(x\right),$ $a_1\left(x\right),$ and $a_0\left(x\right)$ are polynomials with no common factors, then the point $x_0$ is a singular point of the second-order linear DE with polynomial coefficients \[ a_2\left(x\right)y^{\prime\prime} + a_1\left(x\right)y^\prime+a_0\left(x\right)y=0 \] if and only if $x-x_0$ is a factor of $a_2\left(x\right).$

Procedure for determining whether a singular point of a linear, second-order DE with polynomial coefficients is regular or irregular.

  1. Put the DE $a_2\left(x\right)y^{\prime\prime}+a_1\left(x\right)y^\prime+a_0\left(x\right)y=0$ into the standard form \[ y^{\prime\prime} + \frac{a_1\left(x\right)}{a_2\left(x\right)}y^\prime + \frac{a_0\left(x\right)}{a_2\left(x\right)}y=0 \] where $a_0,$ $a_1,$ and $a_2$ are polynomials.
  2. Completely factor (under the set of complex numbers) the polynomials $a_0,$ $a_1,$ and $a_2$ into $n$ distinct linear factors of the form $x-x_0$ (where $n$ is the degree of the polynomial being factored) and reduce the rational functions $\frac{a_1\left(x\right)}{a_2\left(x\right)}$ and $\frac{a_0\left(x\right)}{a_2\left(x\right)}$ to lowest terms. (Such a factorization is guaranteed by the fundamental theorem of algebra.)
  3. For each factor $x-x_0$ of $a_2,$ $x_0$ is a regular singular point if and only if the factor occurs at most once in the reduced form $\frac{a_1\left(x\right)}{a_2\left(x\right)}y^\prime$ and at most twice in the reduced form of $\frac{a_0\left(x\right)}{a_2\left(x\right)}y$ †.
  4. $x_0$ is an irregular singular point if and only if it is a singular point but is not regular.

† According to Dr. Soash, the factor $x-x_0$ of $a_2(x)$ need not appear in the reduced form of either of these terms for the condition to be satisfied, i.e. for $x_0$ to be a regular singular point. That is to say, he claims that the definition of singularity also applies when $a_2(x),$ $a_1(x),$ and $a_0(x)$ are polynomials with common factors. The book’s definition of singularity for polynomial coefficients does not comment on this case, however, it only defines it for the case when they have no common factors. See bottom of p. 271.

Series Solutions of Linear Equations About a Singular Point (Method of Froebenius)

Procedure:

  1. Identify a regular singular point $x_0.$
  2. Substitute \[ y = \sum_{n=0}^{\infty}{c_nx^{n+r}},\quad y^\prime, \quad y^{\prime\prime} \] into the original DE.
  3. Rewrite to find indicial equation and recurrence relation.

  4. Cases

    Case 1: Roots do not differ by a positive integer \[ \eqalign{ & y_1\left(x\right)=\sum_{n=0}^{\infty}{c_0x^{n+r}},\quad c_0\neq0 \\ & y_2\left(x\right)=\sum_{n=0}^{\infty}{c_1x^{n+r_2}},\quad c_1\neq0 } \]

    Case 2: Roots differ by a positive integer, $\left(\left|r_1-r_2\right|\in N\right)$ \[ \eqalign{ & y_1\left(x\right)=\sum_{n=0}^{\infty}{c_nx^{n+r}},\quad c_0\neq1 \\ & y_2\left(x\right)=cy_1\left(x\right)\ln{x}+\sum_{n=0}^{\infty}{b_nx^{n+r_2}},\quad b_0\neq0 } \]

    Case 3: Equal indicial roots $\left(r_1=r_2\right)$ \[ \eqalign{ & y_1\left(x\right)=\sum_{n=0}^{\infty}{c_nx^{n+r_1}},\quad c_0\neq0 \\ & y_2\left(x\right)=y_1\left(x\right)\ln{x}+\sum_{n=0}^{\infty}{b_nx^{n+r_1}} } \]

  5. Use recurrence relation to determine the $c_n\mathrm{'s}$ for the $\sum\mathrm{'s}.$