Chapter 1 – The Basic Terminology of Differential Equations

1.1 What is a Differential Equation?

Differential Equation (DE) is an equation which contains the derivatives of one or more dependent variables with respect to one or more independet variables.

1.2 ODE vs PDE

Ordinary Differential Equation (ODE): A differential equation that contains the derivative of one or more dependent variables, with respect to only one independent variable. Example of an ODE:

(1) $\begin{equation*}\frac{dx}{dt} + \frac{dy}{dt} = 3x + 2y,\end{equation*}$

where $x$ and $y$ are the dependent variables and $t$ is the independent variable.

Partial Differential Equation (PDE): A differential equation that contains the derivative of one or more dependent variables, with respect to more than one independent variables. Example of a PDE:

(2) $\begin{equation*}\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x},\end{equation*}$

where $u$ and $v$ are dependent variables, and $x$ and $y$ are independent variables.

1.3 The Order of a DE

The order of a differential equation is the order of the highest derivative in the equation. A second-order PDE and a third-order ODE are shown below.

(3) $\begin{equation*}\frac{\partial^2 u}{\partial x^2} + \left( \frac{\partial v}{\partial y} \right)^6 = 0, \quad \quad \quad \quad 5 \frac{d^3 y}{d t^3} + 3 \frac{d^2 x}{d t^2} = 0.\end{equation*}$

1.4 Different Forms of a DE

An $n$ th order DE, with $x$ as its independent variable and $y$ as its dependent variable, can be expressed in the general form as:

(4) $\begin{equation*}F(x, y, y^\prime, …, y^{(n)}) = 0,\end{equation*}$

and in the normal form as:

(5) $\begin{equation*}\frac{d^n y}{d x^n} = f \left( x, y, y^\prime, …, y^{(n-1)} \right).\end{equation*}$

1.5 Linear vs Non-Linear ODE

An $n$ th order DE is linear in the variable $y$ if $F$ be linear in $y$ and all its derivatives. Being linear in a $y$ means that the degree of $y$ and all its derivatives must be one, and their coefficients cannot be a function of $y$ or its derivatives. A non-linear DE is simply one that is not linear. More examples for clarification:

1.6 Explicit and Implicit Solutions

A solution in which the dependent variable is expressed solely in terms of the independent variable and constants is said to be an explicit solution.

1.7 Families of Solutions

When solving an n $^{th}$ -order differential equation, $F(x,y,y^\prime,...,y^{(n)})=0$ , we seek an n-parameter family of solutions $G(x,y,c_1,c_2,...,c_n)=0$ . For example, the one-parameter family $y=cx-x \cos x$ is an explicit solution of the linear first-order equation $xy^\prime - y =x^2 \sin x$ .

1.8 Initial-Value Problems

We are often interested in problems in which we seek a solution $y(x)$ of a differential equation so that $y(x)$ satisfies prescribed side conditions – that is, conditions that are imposed on the unknown $y(x)$ or on its derivatives.

1.9 Existence of a Unique Solution

Let $R$ be a rectangular region in the $xy$ -plane defined by $a \leq x \leq b, c \leq y \leq d$ , that contains the point $(x_0,y_0)$ in its interior. If $f(x,y)$ and $\frac{\partial f}{\partial y}$ are continuous on $R$ , then there exists some interval $I_0: (x_0 - h, x_0 + h), h>0$ , contained in $[a,b]$ , and a unique function $y(x)$ defined on $I_0$ that is a solution of the initial-value problem.

1.10 A 10-minute Lecture on the Basic Terminology of Differential Equations

1.11 Problems

State the order of the given differential equations and determine whether it is linear or non-linear.

$(a) \quad \quad (1-x) y^{\prime \prime} - 4 x y^\prime + 5y = \cos x$
Answer: $2^{nd}$ -Order Linear Differential Equation. Watch the solution here.

$(b) \quad \quad \frac{d^2 u}{d r^2} + \frac{du}{dr} + u = \cos (r + u)$
Answer: $2^{nd}$ -Order Non-Linear Differential Equation. Watch the solution here.

$(c) \quad \quad \frac{d^2 y}{dx^2} = \sqrt{1 + \left( \frac{dy}{dx} \right)^2}$
Answer: $2^{nd}$ -Order Non-Linear Differential Equation. Watch the solution here.

Reference

Dennis G. Zill. Advanced Engineering Mathematics, $6^{th}$ edition. Jones $&$ Bartlett Learning. 2016.