Differential Equations Series Solutions and Special Functions Flashcards

24 sample cards from the Series Solutions and Special Functions deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

2. 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.

3. State the recurrence relation connecting Bessel functions of adjacent orders.

   $$J_{\nu-1}(x) + J_{\nu+1}(x) = \frac{2\nu}{x} J_\nu(x)$$

4. 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 $\dfrac{d}{dx}\left[x^{-\nu} J_\nu(x)\right] = -x^{-\nu} J_{\nu+1}(x)$.

5. What is the derivative of $J_0(x)$ in terms of $J_1(x)$?

   $$J_0'(x) = -J_1(x)$$

6. 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 $J_{-1/2}(x) = \sqrt{\dfrac{2}{\pi x}}\,\cos x$.

7. 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 $w(x) = x$.

8. 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 $J_\nu$.

9. 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).

10. 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.

11. 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.

12. 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 $P_n$).

13. 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 $P_n(-1) = (-1)^n$.

14. List the first four Legendre polynomials $P_0, P_1, P_2, P_3$.

    $$P_0(x) = 1, \quad P_1(x) = x, \quad P_2(x) = \tfrac{1}{2}(3x^2 - 1), \quad P_3(x) = \tfrac{1}{2}(5x^3 - 3x)$$

15. State Rodrigues' formula for the Legendre polynomials.

    $$P_n(x) = \frac{1}{2^n\, n!}\,\frac{d^n}{dx^n}\big[(x^2 - 1)^n\big]$$

16. Give the generating function for the Legendre polynomials.

    $$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n=0}^{\infty} P_n(x)\, t^n, \qquad |t| < 1$$

17. State Bonnet's recurrence relation for Legendre polynomials.

    $$(n+1) P_{n+1}(x) = (2n+1)\, x\, P_n(x) - n\, P_{n-1}(x)$$

18. 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 $n$.

19. 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 $w(x)=1$.

20. Write the Legendre (Fourier-Legendre) series expansion of a function $f(x)$ on $[-1,1]$ and its coefficients.

    $$f(x) = \sum_{n=0}^{\infty} c_n P_n(x), \qquad c_n = \frac{2n+1}{2}\int_{-1}^{1} f(x) P_n(x)\, dx$$

21. 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.

22. 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 $x=\pm 1$.

23. 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: $y = c_1 P_n + c_2 Q_n$.

24. 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.

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