Ordinary Differential Equations and Special Functions

Frobenius series solutions

A point x_0 is a regular singular point of y''+p(x)y'+q(x)y=0 when (x-x_0)p(x) and (x-x_0)^2q(x) are both analytic at x_0. Otherwise it is an irregular singular point. Near a regular singular point the ordinary power-series method may fail, so the Frobenius method looks for solutions of the form y=\sum_{n\ge 0}a_n(x-x_0)^{n+r} with a_0\ne 0. Substituting this series and collecting the lowest power of (x-x_0) gives the indicial equation r(r-1)+p_0r+q_0=0, where p_0=\lim_{x\to x_0}(x-x_0)p(x) and q_0=\lim_{x\to x_0}(x-x_0)^2q(x). The roots of this quadratic decide the form of the solutions. For example, x^2y''+x\,y'+x^2y=0 has r^2=0 as its indicial equation, so the Bessel equation of order zero has the double root r=0. Key terms:

• regular singular point — a singular point at which the limits p_0 and q_0 both exist and are finite. The test must be applied after dividing by the leading coefficient, because p and q must be the normalized coefficients. For example, (x-1)^2y''+(x-1)y'+y=0 is regular at x=1. In contrast, (x-1)^2y''+y'+y=0 is irregular there, since (x-1)p(x)=\frac{1}{x-1} is not analytic at 1.
• indicial equation — the quadratic in r obtained from the lowest-order terms. For x^2y''+3xy'+y=0 it reads r(r-1)+3r+1=(r+1)^2=0, giving a double root r=-1.
• difference of roots — if the two roots differ by a non-integer, two independent Frobenius series exist. If they differ by an integer, the series for the smaller root may break down and a logarithmic term may be needed. The absolute difference of the roots is therefore a quantity to compute before choosing the solution form.

The key rule is that Frobenius solutions are power series multiplied by (x-x_0)^r, and the roots of the indicial equation determine how many independent solutions of that form exist.

Sturm-Liouville eigenvalue problems

A Sturm-Liouville problem has the form (p(x)y')'+q(x)y+\lambda w(x)y=0 on [a,b], together with boundary conditions at the endpoints such as y(a)=y(b)=0 or a Robin-type condition that mixes y and y'. For such problems the eigenvalues are real, and eigenfunctions belonging to distinct eigenvalues are orthogonal with respect to the weight w(x). Key points:

• self-adjoint form — an equation written as y''+P(x)y'+\dots=0 is not yet in Sturm-Liouville form. Multiplying by the integrating factor \mu(x)=\exp\!\left(\int P\,dx\right) puts it in that form, and \mu then serves as both p and w. For example, y''+2y'+\lambda y=0 becomes (e^{2x}y')'+\lambda e^{2x}y=0, so the weight is w=e^{2x}.
• eigenvalue computation — for \lambda=k^2>0 the general solution is a combination of sines and cosines. Robin-type conditions lead to transcendental equations for k (equations involving k and a tangent), so the eigenvalues need not be perfect squares.
• mixed boundary conditions — when the conditions link values and derivatives at both ends, eigenvalues of both signs can occur. The cases \lambda=0, \lambda>0 and \lambda<0 must all be checked.

A standard model case is y''+\lambda y=0 with y(0)=y(\pi)=0, which gives \lambda=n^2 and eigenfunctions \sin nx for n=1,2,\dots

Laplace transform method for ODEs

The Laplace transform turns a linear ODE with constant coefficients into an algebraic equation in s. Using \mathcal L\{y'\}=sY-y(0) and \mathcal L\{y''\}=s^2Y-sy(0)-y'(0), the equation y''+ay'+by=f(t) with zero initial data becomes Y(s)=\dfrac{F(s)}{s^2+as+b}. Solving for Y and inverting is the core of the method. Key points:

• convolution theorem — the inverse transform of a product F(s)G(s) is \int_0^t f(\tau)g(t-\tau)\,d\tau. If the denominator is a repeated factor (s+c)^2, the kernel is te^{-ct}, because \mathcal L^{-1}\{1/(s+c)^2\}=te^{-ct}. The extra factor t is the signature of a repeated root.
• unit step and piecewise forcing — a forcing that changes form at some time is written with u(t-a), the unit step. Then \mathcal L\{g(t-a)u(t-a)\}=e^{-as}G(s). For example, the function equal to t on [0,1) and to 1 for t\ge 1 equals t-(t-1)u(t-1).
• Dirac delta forcing — \mathcal L\{\delta(t-a)\}=e^{-as}. For an equation y''+py'+qy=\delta(x-a) with zero initial data, y stays continuous at x=a while y' jumps by the reciprocal of the leading coefficient at that point. Integrating the equation across the impulse is the quickest way to find this jump.

Stability of critical points in linear systems

For a linear system X'=AX, the origin is a critical point, and its type is read from the eigenvalues of A. Real eigenvalues of the same sign give a node, which is asymptotically stable when both are negative and unstable when both are positive. Eigenvalues of opposite sign give a saddle, which is always unstable. Complex eigenvalues \alpha\pm i\beta give a spiral when \beta\ne 0, which is asymptotically stable when \alpha<0. When \alpha=0 the critical point is a center: it is stable but not asymptotically stable. Key points:

• node — for a second-order equation y''+2\alpha y'+\beta^2y=0 with \alpha>\beta>0, the characteristic roots are real and negative, so the origin is an asymptotically stable node.
• coupled systems and decoupling — a system such as x'=y, y'=x can be decoupled by taking the sum x+y and the difference x-y. Each combination satisfies a scalar first-order equation. A solution decays only if its component along the growing direction vanishes, which forces a specific relation between the initial values.
• stability versus asymptotic stability — a conservation law can give stability without decay. If M^{T}=-M (skew-symmetric), then \|X(t)\| is constant along solutions, so the origin is stable. This holds for every such M, regardless of its rank, but it is not asymptotic stability in general.

Legendre polynomial orthogonality

The Legendre polynomials P_n are orthogonal on [-1,1] with respect to the weight function 1. Their norms satisfy \int_{-1}^{1}P_m(x)P_n(x)\,dx=\dfrac{2}{2n+1}\delta_{mn}, so an orthonormal family is obtained by taking \phi_n=C_nP_n with |C_n|=\sqrt{\dfrac{2n+1}{2}}. The sign of C_n is not fixed by orthonormality alone, so values at \pm1 depend on the chosen sign convention. Key points:

• values at endpoints — P_n(1)=1 and P_n(-1)=(-1)^n for every n. Thus a scaled polynomial C_nP_n takes the value \pm C_n at x=\pm 1, depending on the parity of n.
• Legendre equation solutions — a polynomial solution of the Legendre equation of degree k is a constant multiple of P_k. Since the equation fixes the form of the solution, the norms of such solutions are determined by the normalization at x=1, and sums of norms can be telescoped when consecutive degrees are involved.
• orthogonal complement of lower degrees — a polynomial of degree n that is orthogonal to every polynomial of degree less than n is a constant multiple of P_n. The even or odd parity of the polynomial often reduces the orthogonality conditions to a few equations.

Existence and uniqueness of ODE solutions

The Picard-Lindelöf theorem states that if f(x,y) is continuous and \partial f/\partial y is continuous near (x_0,y_0), then the initial value problem y'=f(x,y), y(x_0)=y_0 has a unique solution on some interval around x_0. The theorem is local: it does not by itself say how far the solution extends. Key points:

• global existence — a solution extends until it blows up or leaves the domain of f. If f is bounded in a suitable sense, for instance |f|\le M on the whole plane, then |y(x)-y_0|\le M|x-x_0|, so no blow-up occurs and the solution exists on the whole real line. Uniqueness and global existence are separate questions.
• uniqueness does not imply boundedness — a solution can be unique on all of \mathbb R and still be unbounded. If g is smooth and satisfies 1<g(y)\le 2, then y'\ge 1 on t\ge 0, so y(t)\to\infty as t\to\infty even though the solution is unique.
• failure of uniqueness — for y'=\sqrt{|y|} with y(0)=0, the zero solution and the solutions that leave zero at a later time all satisfy the same initial value. Here \partial f/\partial y is unbounded at y=0, which breaks the hypothesis of the theorem.

Bessel function recurrence relations

Bessel functions J_\alpha satisfy two basic recurrence relations: J_{\alpha-1}(x)+J_{\alpha+1}(x)=\dfrac{2\alpha}{x}J_\alpha(x) and J_{\alpha-1}(x)-J_{\alpha+1}(x)=2J_\alpha'(x). These relations let every J_\alpha be expressed through two consecutive orders, and they are the standard tool for evaluating Bessel functions at specific points. Key points:

• half-integer orders — J_{1/2}(x)=\sqrt{\dfrac{2}{\pi x}}\sin x and J_{-1/2}(x)=\sqrt{\dfrac{2}{\pi x}}\cos x are elementary. Applying the first recurrence with \alpha=\tfrac12 gives J_{3/2}(x)=\dfrac{1}{x}J_{1/2}(x)-J_{-1/2}(x), which is \sqrt{\dfrac{2}{\pi x}}\left(\dfrac{\sin x}{x}-\cos x\right).
• Laplace transform of J_0 — the transform Y(s)=\mathcal L\{J_0(x)\} equals (s^2+1)^{-1/2} for s>0. Differentiating this expression gives a first-order ODE for Y with a coefficient involving s/(s^2+1), which is how the Laplace transform of a Bessel function is connected to the derivative rule \mathcal L\{xf(x)\}=-\dfrac{d}{ds}F(s).
• check at a point — at x=\pi the sine term vanishes and the cosine term equals -1, so the half-integer formulas give simple closed values. Such checks are a quick way to avoid sign errors.