# 0x046 Differential Equation

## 1. Linear ODE

Definition (n-th order ODE) An $$n$$-th order differential equation is said to be linear if it can be written in the form

$y^{(n)}+p_1(x)y^{(n-1)}+ ... + p_n(x)y = f(x)$

For convenience, we consider the following form

$P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+ ... + P_n(x)y = F(x)$

where the left side can be abbreviated as $$Ly$$

Theorem (initial value problem) Suppose $$Ly=F$$ is normal (i.e: $$P_0$$ has nozero and $$P_0, ..., P_n, F$$ are continuous) on $$(a,b)$$, let $$x_0$$ be a point at $$(a,b)$$, and $$k_0, ..., k_{n-1}$$ be arbitrary number, then the initial value problem

$Ly=F, y(x_0)=k_0, ..., y^{(n-1)}(x_0) = k_{n-1}$

has a unique solution on $$(a,b)$$

## 2. Laplace Transform

Definition (Laplace transform) Suppose a function $$f(x)$$ is piecewise-continuous, then the Laplace transform of $$f(t)$$, denoted $$\mathcal{L}\{ f(t) \}$$, is defined as

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

For the sake of convenience, we also use

$F(s) = \mathcal{L} \{ f(t) \}$

Lemma (linearity)

$\mathcal{L} \{ af(t) + bg(t) \} = aF(s)+bG(s)$

Lemma (derivatives) Suppose $$f', ..., f^{(n-1)}$$ are all continuous function and $$f^{(n)}$$ is a piecewise continuous function, then high-order derivatives can be transformed into a linear function

$\mathcal{L}(f^{(n)}) = s^n F(s) - s^{n-1} f(0) - s^{n-2}f'(0) - ... - f^{(n-1)}(0)$

Lemma (convolution) Laplace transform changes convolution into multiplication

$\mathcal{L}(f*g) = F(s)G(s)$

## 3. Linear PDE

2nd order linear PDF can be calssified as either elliptic, hyperbolic or parabolic

$Au_{xx} + 2B u_{xy} + Cu_{yy} + Du_x + Eu_y + Fu + G =00$

by checking sign of $$B^-AC$$

### 3.1. Laplace Equation

Laplace is the simplest example of elliptic PDF

$\nabla^2 u = 0$

or equivalently

$\Delta u = 0$

Its solution is called harmonic functions

### 1-dimension

A trivial case: suppose $$u$$ only depends on $$x$$

$\frac{d^2u}{dx^2} = 0 \implies u(x) = mx + b$

$$u$$ can be fully determined by boundary conditions (e.g: $$(u(0), u(1))$$)

An observations that can be generalized to higher dimension here is

• $$u(x)$$ is the average of $$u(x+a)$$ and $$u(x-a)$$
• no local max or min

#### 3.1.1. 2-dimension

In two dimension:

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

The real and imaginary parts of a complex analytic function both satisfy the Laplae equation. (which can be proved easily using Cauchy-Riemann)

Consider a power series expansion of $$f$$ inside a circle of radius $$R$$

$f(z) = \sum_{n=0}^\infty c_n z^n$

using polar coordinate, we know the real part of $$f(z)$$ is

$a_n r^n \cos(n\theta) - b_n r^n \sin(n\theta)$

which is the solution to the two dimensional Laplace equation, the coeeficient $$a_n, b_n$$ can be determined from the boundary

### 3.2. Heat Equation

Heat equation is an example of the parabolic equation.

### 3.3. Wave Equation

Wave equation is an example of the hyperbolic equation.

By a linear change of variables, any equation of the form

$A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + ... = 0$

with

$B^2 - AC > 0$

This is called the hyperbolic partial differential equation, which can be transformed to the wave equation, ignoring the lower order terms

## 4. Reference

[1] Elementar Elementary Differential E ential Equations with Boundar quations with Boundary Value Problems