Differential Equations Series Solutions and Special Functions Syllabus

Series Solutions and Special Functions syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Series Solutions and Special Functions in Differential Equations, not a summary of it.

1. 1

   Power Series Solutions

   3 topics

   • Review of Power Series
   • Solutions About Ordinary Points
   • Recurrence Relations
2. 2

   Solutions About Singular Points

   2 topics

   • Regular and Irregular Singular Points
   • Method of Frobenius
3. 3

   Special Functions

   3 topics

   • Bessel's Equation and Bessel Functions
   • Legendre's Equation and Legendre Polynomials
   • Properties and Orthogonality

Series Solutions and Special Functions flashcards for Differential Equations

19 of 50 cards from the Series Solutions and Special Functions deck — real questions with worked answers.

1. What is the definition of a power series centered at $x_0$?

   A power series about $x_0$ is an infinite series of the form $$\sum_{n=0}^{\infty} a_n (x-x_0)^n = a_0 + a_1(x-x_0) + a_2(x-x_0)^2 + \cdots$$ where the $a_n$ are constant coefficients.

2. What does the radius of convergence $R$ of a power series tell you, and how is it often found?

   The series $\sum a_n (x-x_0)^n$ converges absolutely for $|x-x_0| < R$ and diverges for $|x-x_0| > R$. It is often found via the ratio test: $$R = \lim_{n\to\infty} \left| \frac{a_n}{a_{n+1}} \right|$$

3. When is a function $f(x)$ said to be analytic at a point $x_0$?

   $f$ is analytic at $x_0$ if it can be represented by a power series $\sum_{n=0}^{\infty} a_n (x-x_0)^n$ with positive radius of convergence in some neighborhood of $x_0$; equivalently, its Taylor series converges to $f$ near $x_0$.

4. State the Taylor series formula for the coefficients $a_n$ of a function $f$ analytic at $x_0$.

   $$a_n = \frac{f^{(n)}(x_0)}{n!}, \qquad f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$

5. How do you differentiate a power series $y = \sum_{n=0}^{\infty} a_n x^n$ term by term?

   $$y' = \sum_{n=1}^{\infty} n a_n x^{n-1}, \qquad y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}$$ valid within the radius of convergence.

6. What is the Identity (Vanishing) Property used when solving ODEs by power series?

   If $\sum_{n=0}^{\infty} c_n (x-x_0)^n = 0$ for all $x$ in an interval, then every coefficient must vanish: $c_n = 0$ for all $n$. This lets you set each coefficient to zero to get recurrence relations.

7. For the standard-form ODE $y'' + P(x)y' + Q(x)y = 0$, what defines an ordinary point $x_0$?

   $x_0$ is an ordinary point if both $P(x)$ and $Q(x)$ are analytic at $x_0$. Otherwise $x_0$ is a singular point.

8. State the existence theorem for power series solutions about an ordinary point.

   If $x_0$ is an ordinary point of $y'' + P(x)y' + Q(x)y = 0$, then there exist two linearly independent analytic solutions of the form $\sum_{n=0}^{\infty} a_n (x-x_0)^n$, and their radius of convergence is at least the distance from $x_0$ to the nearest singular point (in the complex plane).

9. How many arbitrary constants appear in the general power series solution about an ordinary point of a 2nd-order ODE, and which coefficients are they?

   Two arbitrary constants, typically $a_0$ and $a_1$, which correspond to the initial values $y(x_0)$ and $y'(x_0)$. All higher coefficients are determined in terms of them by the recurrence relation.

10. What is a recurrence relation in the context of power series solutions?

    An equation obtained by substituting the series into the ODE and equating coefficients, expressing a coefficient $a_{n+k}$ in terms of one or more earlier coefficients (e.g. $a_n$), which generates all coefficients from the initial ones $a_0, a_1$.

11. Derive the recurrence relation for $y'' - x y = 0$ (Airy's equation) about $x_0 = 0$.

    Substituting $y = \sum a_n x^n$ gives $\sum_{n=2} n(n-1)a_n x^{n-2} - \sum a_n x^{n+1} = 0$, leading to $a_2 = 0$ and $$a_{n+2} = \frac{a_{n-1}}{(n+2)(n+1)}, \quad n \geq 1$$

12. For the ODE $y'' + y = 0$ solved by power series about $x_0=0$, what is the recurrence relation and the resulting solutions?

    The recurrence is $a_{n+2} = -\dfrac{a_n}{(n+2)(n+1)}$. This yields $y = a_0 \cos x + a_1 \sin x$, matching the known cosine and sine series.

13. In the standard form $y'' + P(x)y' + Q(x)y = 0$, what makes $x_0$ a regular singular point?

    $x_0$ is a regular singular point if it is a singular point but both $(x-x_0)P(x)$ and $(x-x_0)^2 Q(x)$ are analytic at $x_0$.

14. What is an irregular singular point?

    A singular point $x_0$ is irregular if it is not regular, i.e. at least one of $(x-x_0)P(x)$ or $(x-x_0)^2 Q(x)$ fails to be analytic at $x_0$.

15. Classify $x=0$ for the equation $x^2 y'' + x y' + (x^2 - \nu^2) y = 0$ (Bessel's equation).

    Writing in standard form: $P(x)=\frac{1}{x}$, $Q(x)=\frac{x^2-\nu^2}{x^2}$. Since $xP(x)=1$ and $x^2 Q(x)=x^2-\nu^2$ are both analytic at $0$, $x=0$ is a regular singular point.

16. Classify $x=0$ for the equation $x^3 y'' + y = 0$.

    Here $P=0$, $Q = 1/x^3$. Then $x^2 Q(x) = 1/x$ is not analytic at $0$, so $x=0$ is an irregular singular point.

17. What form of solution does the Method of Frobenius seek about a regular singular point $x_0=0$?

    $$y = x^{r} \sum_{n=0}^{\infty} a_n x^{n} = \sum_{n=0}^{\infty} a_n x^{n+r}, \qquad a_0 \neq 0$$ where the exponent $r$ (possibly non-integer or complex) is determined by the indicial equation.

18. What is the indicial equation and how is it obtained?

    Write $x P(x) = \sum p_n x^n$ and $x^2 Q(x) = \sum q_n x^n$ with leading terms $p_0, q_0$. The indicial equation is $$r(r-1) + p_0 r + q_0 = 0$$ obtained from the lowest power of $x$ ($a_0 \neq 0$). Its roots $r_1, r_2$ are the exponents at the singularity.

19. State the three cases of the Frobenius method based on the indicial roots $r_1 \geq r_2$ (real).

    Case 1: $r_1 - r_2$ not an integer — two independent Frobenius solutions with exponents $r_1$ and $r_2$. Case 2: $r_1 = r_2$ (equal roots) — a second solution contains a logarithmic term. Case 3: $r_1 - r_2$ a positive integer — second solution may or may not contain a logarithmic term.

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