ODE-09

Special Topics

Special Equations and Applications

Explore important special differential equations that arise in physics and engineering: Bessel and Legendre equations, power series solutions, Laplace transforms, and real-world applications.

Learning Objectives

Solve ODEs using power series methods
Classify ordinary and singular points
Apply the Frobenius method
Understand Bessel functions and their properties
Work with Legendre polynomials
Apply Laplace transforms to solve ODEs
Model physical systems with ODEs
Connect special functions to applications

1. Power Series Solutions

When explicit solutions are not available, power series provide a systematic way to construct solutions term by term.

Definition 9.1: Ordinary and Singular Points

For the equation $y'' + p(x)y' + q(x)y = 0$ :

$x_0$ is an ordinary point if $p(x)$ and $q(x)$ are analytic at $x_0$
$x_0$ is a regular singular point if $(x-x_0)p(x)$ and $(x-x_0)^2 q(x)$ are analytic at $x_0$
Otherwise, $x_0$ is an irregular singular point

Theorem 9.1: Power Series at Ordinary Points

If $x_0$ is an ordinary point of $y'' + p(x)y' + q(x)y = 0$ , then there exist two linearly independent solutions of the form:

y = \sum_{n=0}^{\infty} a_n (x - x_0)^n

The series converges at least in the largest disk centered at $x_0$ where $p$ and $q$ are analytic.

Algorithm 9.1: Power Series Method

Input: ODE $y'' + p(x)y' + q(x)y = 0$ , ordinary point $x_0$

Output: Power series solution

1. Assume $y = \sum a_n (x - x_0)^n$

2. Compute $y' = \sum n a_n (x - x_0)^,{n-1}$

3. Compute $y'' = \sum n(n-1) a_n (x - x_0)^,{n-2}$

4. Substitute into ODE and collect like powers of $(x - x_0)$

5. Set each coefficient to zero → recurrence relation

6. Solve recurrence for $a_n$ in terms of $a_0, a_1$

Example 9.1: Power Series Solution

Problem: Find a power series solution of $y'' + y = 0$ about $x = 0$ .

Solution:

Let $y = \sum_,{n=0},^,infty, a_n x^n$ . Then:

y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} = \sum_{m=0}^{\infty} (m+2)(m+1) a_{m+2} x^m

Substituting into $y'' + y = 0$ :

\sum_{n=0}^{\infty} [(n+2)(n+1)a_{n+2} + a_n] x^n = 0

Recurrence: $a_,{n+2}, = -\frac,{a_n}{(n+2)(n+1)}$

With $a_0$ and $a_1$ arbitrary:

a_2 = -\frac{a_0}{2}, \quad a_3 = -\frac{a_1}{6}, \quad a_4 = \frac{a_0}{24}, \quad a_5 = \frac{a_1}{120}, \ldots

This gives $y = a_0 \cos x + a_1 \sin x$ .

2. The Frobenius Method

Theorem 9.2: Frobenius Method

If $x = 0$ is a regular singular point, at least one solution has the form:

y = x^r \sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} a_n x^{n+r}

where $r$ is determined by the indicial equation and $a_0 \neq 0$ .

Definition 9.2: Indicial Equation

For the equation $x^2 y'' + xp(x)y' + q(x)y = 0$ with $p(x) = \sum p_n x^n$ , $q(x) = \sum q_n x^n$ , the indicial equation is:

r(r-1) + p_0 r + q_0 = 0

The roots $r_1, r_2$ determine the exponents at the singular point.

Example 9.2: Frobenius Method Example

Problem: Solve $2xy'' + y' + y = 0$ near $x = 0$ .

Solution:

Standard form: $y'' + \frac,{1}{2x},y' + \frac,{1}{2x},y = 0$

Here $xp(x) = \frac,{1}{2}$ and $x^2 q(x) = \frac,{x}{2}$ are analytic at $x = 0$ , so $x = 0$ is a regular singular point.

Indicial equation: $r(r-1) + \frac,{1}{2},r + 0 = r^2 - \frac,{1}{2},r = r(r - \frac,{1}{2},) = 0$

Roots: $r_1 = \frac,{1}{2},, r_2 = 0$

For $r = \frac,{1}{2}$ , we get a solution involving $x^,{1/2}$ and a convergent series.

3. Bessel's Equation

Definition 9.3: Bessel's Equation

Bessel's equation of order $\nu$ is:

x^2 y'' + xy' + (x^2 - \nu^2)y = 0

Solutions are Bessel functions $J_\nu(x)$ (first kind) and $Y_\nu(x)$ (second kind).

Theorem 9.3: Bessel Function of the First Kind

The Bessel function of the first kind is:

J_\nu(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \, \Gamma(m + \nu + 1)} \left(\frac{x}{2}\right)^{2m + \nu}

For integer $n$ , $J_n(x)$ is bounded at $x = 0$ .

Remark:

Bessel functions appear in: vibrating circular membranes, heat conduction in cylinders, electromagnetic waves in cylindrical waveguides, and quantum mechanics of the hydrogen atom.

4. Legendre's Equation

Definition 9.4: Legendre's Equation

Legendre's equation is:

(1 - x^2)y'' - 2xy' + n(n+1)y = 0

For non-negative integer $n$ , one solution is the Legendre polynomial $P_n(x)$ .

Theorem 9.4: Legendre Polynomials

Legendre polynomials can be defined by Rodrigues' formula:

P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2 - 1)^n

First few: $P_0 = 1, P_1 = x, P_2 = \frac,{1}{2},(3x^2 - 1), P_3 = \frac,{1}{2},(5x^3 - 3x)$

Remark:

Legendre polynomials appear in: spherical harmonics, multipole expansions in electrostatics, and solutions of Laplace's equation in spherical coordinates.

5. Laplace Transform Method

Definition 9.5: Laplace Transform

The Laplace transform of $f(t)$ is:

\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) \, dt

Key Laplace Transform Properties

$\mathcal{L}\{f'\} = sF(s) - f(0)$

$\mathcal{L}\{f''\} = s^2 F(s) - sf(0) - f'(0)$

$\mathcal{L}\{e^{at}f\} = F(s-a)$

$\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$

$\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}$

$\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$

Algorithm 9.2: Solving ODEs with Laplace Transforms

Input: ODE with initial conditions

Output: Solution $y(t)$

1. Take Laplace transform of both sides

2. Substitute initial conditions

3. Solve algebraically for $Y(s)$

4. Take inverse Laplace transform to get $y(t)$

Example 9.3: Laplace Transform Solution

Problem: Solve $y'' + 4y = 0$ , $y(0) = 1$ , $y'(0) = 0$ .

Solution:

Taking Laplace transform:

s^2 Y - s(1) - 0 + 4Y = 0

Y(s^2 + 4) = s \implies Y = \frac{s}{s^2 + 4}

Inverse transform:

y(t) = \mathcal{L}^{-1}\left\{\frac{s}{s^2 + 4}\right\} = \cos(2t)

6. Applications in Physics and Engineering

Example 9.4: Vibrating Circular Membrane

Problem: A circular drum of radius $a$ vibrates. Find the vibration modes.

Solution:

The wave equation in polar coordinates separates into:

r^2 R'' + rR' + (\lambda^2 r^2 - n^2)R = 0

This is Bessel's equation! The bounded solutions are $R(r) = J_n(\lambda r)$ .

The boundary condition $R(a) = 0$ gives $J_n(\lambda a) = 0$ , determining the eigenvalues from the zeros of Bessel functions.

Example 9.5: RLC Circuit

Problem: An RLC circuit has $L\frac,{d^2 q}{dt^2}, + R\frac,{dq}{dt}, + \frac,{q}{C}, = E(t)$ .

Solution:

This is a second-order constant coefficient ODE. The Laplace transform is ideal here:

L(s^2 Q - sq_0 - q_0') + R(sQ - q_0) + \frac{Q}{C} = \mathcal{L}\{E(t)\}

Solving for $Q(s)$ and inverting gives the charge $q(t)$ .

Practice Quiz

Special Equations Quiz

Questions

Correct

Accuracy

The Bessel equation of order

n

is:

Easy

Not attempted

The Legendre equation is:

Easy

Not attempted

A point

x_0

is an ordinary point of

y'' + p(x)y' + q(x)y = 0

if:

Easy

Not attempted

The Frobenius method is used when:

Medium

Not attempted

The indicial equation in the Frobenius method determines:

Medium

Not attempted

The Laplace transform of

f(t)

is defined as:

Medium

Not attempted

The Laplace transform of

f'(t)

is:

Medium

Not attempted

Which physical problem leads to Bessel's equation?

Medium

Not attempted

In the power series solution

y = \sum a_n x^n

, the recurrence relation typically:

Hard

Not attempted

The Airy equation

y'' - xy = 0

has solutions that:

Hard

Not attempted

Frequently Asked Questions

When should I use power series vs. other methods?

Use power series when: (1) the equation has variable coefficients, (2) solutions are not elementary functions, (3) you need local behavior near a point. For constant coefficients or standard forms, use characteristic equations or Laplace transforms instead.

What makes a singular point 'regular'?

At a regular singular point,

(x-x_0)p(x)

and

(x-x_0)^2 q(x)

are analytic. This means the singularity is "mild enough" that Frobenius series still work. Irregular singular points (like those of Airy's equation at infinity) require more sophisticated methods.

Why are Bessel and Legendre functions so important?

They arise naturally when solving PDEs in cylindrical (Bessel) or spherical (Legendre) coordinates. Since many physical problems have this symmetry—drums, pipes, atoms, planets—these functions appear throughout physics and engineering.

When is the Laplace transform most useful?

Laplace transforms excel for: (1) initial value problems with constant coefficients, (2) discontinuous or impulsive forcing (like delta functions), (3) systems with delays, and (4) linear control theory. They convert differential equations to algebra.

How do I find the inverse Laplace transform?

Common methods: (1) table lookup, (2) partial fraction decomposition, (3) completing the square for quadratics, (4) convolution theorem. Most practical problems reduce to combinations of standard transforms.

What is the connection between special functions and orthogonality?

Bessel functions, Legendre polynomials, and other special functions form orthogonal sets with respect to appropriate weight functions. This orthogonality allows Fourier-like expansions and is fundamental to solving boundary value problems.