Note #1
MATH 310
Introduction to Ordinary Di▯erential Equations
CLASSIFICATION OF ORDINARY DIFFERENTIAL EQUATIONS
What is a di▯erential equation? It is an equation that establishes a functional relationship between
one independent variable and one dependent variable. For example, let us assume that the
dependent variable is y and that the independent variable is t. The equation
dy
+ ln(t)y = 1 (1)
dt
establishes a functional relationship y = y(t) for t > 0. How do we ▯nd y(t)? That is the
central question of this course. To discuss di▯erential equations, it is ▯rst necessary to de▯ne
some vocabulary that allows us to classify di▯erential equations.
Order of a Di▯erential Equation
As before, let us assume that the dependent variable is y and that the independent variable is
n n
t. A derivative of order n is usually written in the following manner: d y=dt . The order of a
di▯erential equation is the same as the order of the highest derivative that appears in the equation.
Equation (1) is thus a ▯rst-order equation.
Ordinary vs. Partial Di▯erential Equation
If the di▯erential equation (or equations) contain only one independent variable, then we say that
the di▯erential equation (or equations) are ordinary. If the di▯erential equation (or equations) con-
tain more than one independent variable, then we say that the di▯erential equation (or equations)
are partial. One example of a partial di▯erential equation is the transport equation:
@u @u
@t +

[email protected] = 0 where u = u(x;t) and a = constant: (2)
An example of a ▯rst-order ordinary di▯erential equation was given earlier in (1).
Systems of Ordinary Di▯erential Equations
If were to have a collection of m dependent variables, we would need m di▯erential equations to
completely specify the m dependent variables. If m > 1, we say that we are dealing with a system
of di▯erential equations. For example, the system of two coupled equations
dv + aw = sin(t) where a = constant (3)
dt
and
dw
dt + bv = t + 1 where b = constant: (4)
speci▯es the two functions w(t) and v(t). Because there is only one independent variable, this is
a system of ordinary rather than partial di▯erential equations.
Fall 2013 Page 1/3 MATH 310 Introduction to Ordinary Di▯erential Equations Note #1
Linearity
A function f(t) is said to be linear if it satis▯es the following property:
f(▯t) = ▯f(t) for any constant ▯: (5)
In other words, if we were to make the substitution t ! ▯t in f(t), the result should be the same
as what we would get if we muiltiplied f(t) by ▯. The concept of linearity applies in a slightly
di▯erent way to equations. The test for linearity is de▯ned in terms of the dependent variable
rather than in terms of the independent variable. Suppose that y is the dependent variable and
t is the independent variable. To establish the linearity of a di▯erential equation that speci▯es
y = y(t); we must ▯rst rewrite the di▯erential equation the following form:
▯ n ▯
F t;y;dy;:::;d y = G(t) (6)
dt dtn
where F contains all the terms that are dependent on y and G(t) contains all the te