1550078481-Ordinary_Differential_Equations__Roberts_

Introduction 7

contains partial derivatives, then the differential equation is called a partial

differential equation (PDE). The order of a differential equation, ordinary

or partial, is the order of the highest derivative occuring in the equation.

For example,

dy

(1) dx =cosy

is a first-order, ordinary differential equation. The dependent variable is y

and the independent variable is x- that is, the unknown function is y( x).

The equation

(2) d

3

y (dy)

2

• 
  
  
  
  • x - + x^2 y = tan x
    dx 3 dx
  
  
  
  

is a third-order, ordinary differential equation.

The equation

(3)

(d^4 y )^3 d2 y^5 t

dt4 + ty dt2 - Y = e

is a fourth-order, ordinary differential equation. The dependent variable is y
and the independent variable is t- thus, the unknown function is y(t).

The equation

(4)

oz oz

-+-=z

ax ay

is a first-order, partial differential equation. The dependent variable is z and
the independent variables are x and y. Hence, the unknown function is z(x, y).

The equation

(5)

au

at

is a second-order, partial differential equation. The unknown function is
u(x, y, z, t).

d2y

In calculus, you used the notations dx 2 and y" to represent the second

derivative of y. In the first notation, it is clear that x is the independent
vari able. In the second notation, the prime notation, it is not obvious what the
independent variable is. The first notation is somewhat cumbersome, while
the second is convenient only for writing lower order derivatives. Reading
and writing the tenth derivative using prime notation would be tedious, so