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Differential Equations Series Solutions and Special Functions Syllabus
The Series Solutions and Special Functions syllabus for Differential Equations in full: 3 chapters and 8 topics, every one listed below, plus 50 Series Solutions and Special Functions flashcards with answers.
Series Solutions and Special Functions syllabus: all 3 chapters
Expand any chapter to see its topics and sub-topics.
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Power Series Solutions
3 topics- Review of Power Series
- Solutions About Ordinary Points
- Recurrence Relations
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Solutions About Singular Points
2 topics- Regular and Irregular Singular Points
- Method of Frobenius
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Special Functions
3 topics- Bessel's Equation and Bessel Functions
- Legendre's Equation and Legendre Polynomials
- Properties and Orthogonality
Differential Equations Series Solutions and Special Functions flashcards — 50 questions with answers
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.
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: $$R = \lim_{n\to\infty} \left| \frac{a_n}{a_{n+1}} \right|$$
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 $x_0$.
State the Taylor series formula for the coefficients $a_n$ of a function $f$ analytic at $x_0$.
$$a_n = \frac{f^{(n)}(x_0)}{n!}, \qquad f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$
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.
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.
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.
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).
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.
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 $a_0, a_1$.
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 $$a_{n+2} = \frac{a_{n-1}}{(n+2)(n+1)}, \quad n \geq 1$$
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.
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 $x_0$.
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 $x_0$.
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.
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.
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.
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.
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.
In the equal-roots Frobenius case ($r_1 = r_2 = r$), what is the form of the two linearly independent solutions?
$$y_1 = x^{r}\sum_{n=0}^{\infty} a_n x^n, \qquad y_2 = y_1 \ln x + x^{r}\sum_{n=1}^{\infty} b_n x^n$$
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In the Frobenius case where $r_1 - r_2 = N$ is a positive integer, what is the general form of the second solution?
$$y_1 = x^{r_1}\sum_{n=0}^{\infty} a_n x^n, \qquad y_2 = C\, y_1 \ln x + x^{r_2}\sum_{n=0}^{\infty} b_n x^n$$ where the constant $C$ may turn out to be zero (then no log term).
Write Bessel's differential equation of order $\nu$.
$$x^2 y'' + x y' + (x^2 - \nu^2) y = 0$$ where $\nu \geq 0$ is a real parameter (the order).
What are the indicial roots of Bessel's equation of order $\nu$?
The indicial equation is $r^2 - \nu^2 = 0$, giving roots $r = \nu$ and $r = -\nu$.
Give the series definition of the Bessel function of the first kind $J_\nu(x)$.
$$J_\nu(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\,\Gamma(m+\nu+1)} \left(\frac{x}{2}\right)^{2m+\nu}$$
When is $\{J_\nu, J_{-\nu}\}$ a fundamental (linearly independent) set of solutions to Bessel's equation, and when does it fail?
$J_\nu$ and $J_{-\nu}$ are linearly independent when $\nu$ is not an integer. When $\nu = n$ is an integer, $J_{-n}(x) = (-1)^n J_n(x)$, so they are linearly dependent and a second independent solution is needed.
What is the Bessel function of the second kind $Y_\nu(x)$ (Weber/Neumann function) used for?
$Y_\nu(x)$ provides a second solution linearly independent of $J_\nu$ for all $\nu$ (including integer order). It is defined by $$Y_\nu(x) = \frac{J_\nu(x)\cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)}$$ (with a limit taken for integer $\nu$). It is singular ($\to -\infty$) as $x \to 0^+$.
Write the general solution of Bessel's equation of order $\nu$.
$$y(x) = c_1 J_\nu(x) + c_2 Y_\nu(x)$$ valid for all $\nu \geq 0$, since $J_\nu$ and $Y_\nu$ are always linearly independent.
What is the behavior of $J_0(0)$ and $J_n(0)$ for $n \geq 1$?
$J_0(0) = 1$, while $J_n(0) = 0$ for every integer $n \geq 1$. Every $J_n$ with $n\geq 1$ starts at zero at the origin.
State the recurrence relation connecting Bessel functions of adjacent orders.
$$J_{\nu-1}(x) + J_{\nu+1}(x) = \frac{2\nu}{x} J_\nu(x)$$
State the derivative recurrence relations for Bessel functions.
$$J_{\nu-1}(x) - J_{\nu+1}(x) = 2 J_\nu'(x), \qquad \frac{d}{dx}\left[x^\nu J_\nu(x)\right] = x^\nu J_{\nu-1}(x)$$ and $\dfrac{d}{dx}\left[x^{-\nu} J_\nu(x)\right] = -x^{-\nu} J_{\nu+1}(x)$.
What is the derivative of $J_0(x)$ in terms of $J_1(x)$?
$$J_0'(x) = -J_1(x)$$
Express the half-integer Bessel function $J_{1/2}(x)$ in elementary form.
$$J_{1/2}(x) = \sqrt{\frac{2}{\pi x}}\,\sin x$$ and correspondingly $J_{-1/2}(x) = \sqrt{\dfrac{2}{\pi x}}\,\cos x$.
State the orthogonality relation for Bessel functions $J_\nu$ on $[0,1]$ with respect to weight $x$.
If $\alpha_i, \alpha_j$ are distinct positive roots of $J_\nu$, then $$\int_0^1 x\, J_\nu(\alpha_i x)\, J_\nu(\alpha_j x)\, dx = 0 \quad (i \neq j)$$ The weight function is $w(x) = x$.
What is the normalization integral for Bessel functions (the $i=j$ case of orthogonality)?
$$\int_0^1 x\, \big[J_\nu(\alpha_i x)\big]^2\, dx = \frac{1}{2}\big[J_{\nu+1}(\alpha_i)\big]^2 = \frac{1}{2}\big[J_\nu'(\alpha_i)\big]^2$$ where $\alpha_i$ is a positive zero of $J_\nu$.
Write Legendre's differential equation.
$$(1-x^2) y'' - 2x y' + n(n+1) y = 0$$ where $n$ is a parameter (often a non-negative integer).
Classify the points $x = \pm 1$ and $x = 0$ for Legendre's equation.
$x = 0$ is an ordinary point (so power series solutions exist there), while $x = 1$ and $x = -1$ are regular singular points.
State the recurrence relation for the power series solution of Legendre's equation about $x=0$.
$$a_{k+2} = \frac{(k-n)(k+n+1)}{(k+1)(k+2)}\, a_k$$ Substituting $y=\sum a_k x^k$ into Legendre's equation yields this two-term recurrence.
Why does Legendre's equation have polynomial solutions when $n$ is a non-negative integer?
When $n$ is a non-negative integer, the recurrence factor $(k-n)(k+n+1)$ vanishes at $k=n$, so $a_{n+2}=0$ and the series terminates — producing a polynomial of degree $n$ (the Legendre polynomial $P_n$).
What is the standard normalization used to define the Legendre polynomials $P_n(x)$?
They are normalized so that $$P_n(1) = 1 \quad \text{for every } n.$$ Also $P_n(-1) = (-1)^n$.
List the first four Legendre polynomials $P_0, P_1, P_2, P_3$.
$$P_0(x) = 1, \quad P_1(x) = x, \quad P_2(x) = \tfrac{1}{2}(3x^2 - 1), \quad P_3(x) = \tfrac{1}{2}(5x^3 - 3x)$$
State Rodrigues' formula for the Legendre polynomials.
$$P_n(x) = \frac{1}{2^n\, n!}\,\frac{d^n}{dx^n}\big[(x^2 - 1)^n\big]$$
Give the generating function for the Legendre polynomials.
$$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n=0}^{\infty} P_n(x)\, t^n, \qquad |t| < 1$$
State Bonnet's recurrence relation for Legendre polynomials.
$$(n+1) P_{n+1}(x) = (2n+1)\, x\, P_n(x) - n\, P_{n-1}(x)$$
What is the parity (even/odd symmetry) of $P_n(x)$?
$P_n(x)$ has the parity of $n$: $$P_n(-x) = (-1)^n P_n(x)$$ so $P_n$ is an even function for even $n$ and an odd function for odd $n$.
State the orthogonality relation for Legendre polynomials on $[-1,1]$.
$$\int_{-1}^{1} P_m(x)\, P_n(x)\, dx = \begin{cases} 0, & m \neq n \\[4pt] \dfrac{2}{2n+1}, & m = n \end{cases}$$ The weight function is $w(x)=1$.
Write the Legendre (Fourier-Legendre) series expansion of a function $f(x)$ on $[-1,1]$ and its coefficients.
$$f(x) = \sum_{n=0}^{\infty} c_n P_n(x), \qquad c_n = \frac{2n+1}{2}\int_{-1}^{1} f(x) P_n(x)\, dx$$
In Sturm-Liouville theory, what general property guarantees the orthogonality of eigenfunctions like $P_n$ and $J_\nu$?
Both arise as eigenfunctions of a self-adjoint (Sturm-Liouville) problem $\frac{d}{dx}[p(x)y'] + [q(x) + \lambda w(x)]y = 0$ with suitable boundary conditions. Eigenfunctions for distinct eigenvalues are orthogonal with respect to the weight $w(x)$ — giving weight $1$ for Legendre and weight $x$ for Bessel.
Compare the singular-point structure of Bessel's equation and Legendre's equation at their key points.
Bessel's equation has a regular singular point at $x=0$ (solved by Frobenius), whereas Legendre's equation has an ordinary point at $x=0$ (solved by ordinary power series) and regular singular points at $x=\pm 1$.
What is the relationship between the two Legendre solutions $P_n(x)$ and $Q_n(x)$?
$P_n(x)$ is the polynomial solution, bounded on $[-1,1]$. $Q_n(x)$ is the Legendre function of the second kind — a second, linearly independent solution that is unbounded (logarithmically singular) at $x = \pm 1$. General solution: $y = c_1 P_n + c_2 Q_n$.
Summarize the overall strategy for choosing a series method at a point $x_0$ of a linear 2nd-order ODE.
First classify $x_0$: if ordinary, use an ordinary power series $\sum a_n(x-x_0)^n$ (two analytic solutions guaranteed). If a regular singular point, use the Method of Frobenius $\sum a_n (x-x_0)^{n+r}$ with the indicial equation. If irregular singular, standard series methods fail and other techniques are required.
Differential Equations Series Solutions and Special Functions FAQ
What is in the Differential Equations Series Solutions and Special Functions syllabus?
Differential Equations Series Solutions and Special Functions is split into 3 chapters: Power Series Solutions, Solutions About Singular Points and Special Functions. Together they contain 8 topics.
How many chapters are in Differential Equations Series Solutions and Special Functions?
3 chapters, containing 8 topics. The largest chapter is Power Series Solutions, with 3 topics.
Are there flashcards for Differential Equations Series Solutions and Special Functions?
Yes — a 50-card Series Solutions and Special Functions deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.