Introduction
Ordinary Differential Equations
Dr. Marco A Roque Sol
12/01/2015
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Table of contents
Introduction
First Order Differential Equations
Second Order Differential Equations
Higher Order Differential Equations
Series
The Laplace Transform
System of First Order Linear Differential Equations
Nonlinear Differential Equations
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Definitions
Classification
Introduction
Figure: Sophus Lie.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Definitions
Classification
Definitions
Differential Equations
A differential equation is any equation which contains
derivatives, either ordinary or partial derivatives.
There many situations where we ca find such an object. Thus, for
instance in the case of the study of Classical Mechanics in Physics,
if an object of mass m is moving with acceleration a and being
acted on with force F then Newtons Second Law tells us.
F = ma
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Definitions
Classification
Definitions
To see that this is in fact a differential equation. First, remember
that we can rewrite the acceleration, a, in one of two ways.
dv
d 2u
or a = 2
dt
dt
Where v is the velocity of the object and u is the position function
of the object at any time t. We should also remember at this point
that the force, F may also be a function of time, velocity, and/or
position.
a=
m
dv
= F (t, v ) or
dt
Dr. Marco A Roque Sol
m
d 2u
= F (t, u, v )
dt 2
Ordinary Differential Equations
Introduction
Definitions
Classification
Definitions
More examples of differential equations
2y 00 + 3y 0 − 5y = 0
dy
d 2y
+ y 3 e −y = 0
cos(y ) 2 − (1 + y )
dx
dx
y (4) + 5y 000 − 4y 00 + y = sin(x)
∂2u
∂u
a2 2 =
∂x
∂t
2
∂ u
∂2u
a2 2 = 2
∂x
∂t
3
∂u
∂u
=1+
∂ 2 x∂t
∂y
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Definitions
Classification
Definitions
Solution
A solution to a differential equation on an interval α < t < β is
any function y (t) which satisfies the differential equation in
question on the interval. It is important to note that solutions are
often accompanied by intervals and these intervals can impart
some important information about the solution.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Definitions
Classification
Classification
Order
The order of a differential equation is the largest derivative present
in the differential equation. The equation
dv
m
= F (t, v )
dt
is a first order differential equation, the equations
d 2u
m 2 = F (t, u, v )
dt
2y 00 + 3y 0 − 5y = 0
d 2y
dy
cos(y ) 2 − (1 + y )
+ y 3 e −y = 0
dx
dx
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Definitions
Classification
Classification
∂2u
∂u
=
∂x 2
∂t
2
∂ u
∂2u
a2 2 = 2
∂x
∂t
are second order differential equations, the equation
∂u 3
∂u
=1+
∂ 2 x∂t
∂y
is a third order differential equation and finally, the equation
a2
y (4) + 5y 000 − 4y 00 + y = sin(x)
is a fourth order differential equation.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Introduction
Definitions
Classification
Classification
Note that the order does not depend on whether or not you’ve got
ordinary or partial derivatives in the differential equation.
As you will see most of the solution techniques for second order
differential equations can be easily (and naturally) extended to
higher order differential equations .
Ordinary and Partial Differential Equations
A differential equation is called an ordinary differential equation,
abbreviated by ODE, if it has ordinary derivatives in it. Likewise, a
differential equation is called a partial differential equation,
abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol
Ordinary Differential Equations