We show the structural similarities between the
linear algebraic equation and the linear differential
equation, and introduce the concept of
superposition of solutions for linear homogeneous
equations.
We then present the important nonhomogeneous
principle, which gives the form of the
solutions for the general nonhomogeneous linear
equation, either algebraic or differential , as
the sum of the homogeneous solutions plus any
particular solution.
What is Linear?
An algebraic equation 
is
linear if it is of the form
where 
and C are
constants. If
C = 0, the algebraic equation is said to be
homogeneous.
Example 1 Examine the fol lowing equations
to de termine which ones are linear
Example 2 The equation 5x + 4y - 3z = 4
is a linear equation that describes a plane in
3-space.
.
Example 3 A student has a monthly entertainment
budget of $50. The expected costs
are as follows: $8 for a movie ticket, $2 for
a video rental, $7 for a new paperback book
and $12 for a compact disc. This information
is represented by the linear equation
8m+2v +7b+12d = 50,
where m, v, b, and d represent the number of
movie tickets, videos, books and CDs, respectively,
that the student can buy or rent in a
month if the entire budget is spent.
A differential equation 
is linear if it is of the form
where all the functions of t are assumed to
be defined over some common interval I. If
f(t) = 0 over the interval I, the differential
equation is said to be homogeneous.
In particular, the general first- order linear differential
equation can be written as
y' +p(t)y = f(t).
The general second order linear differential equation
is
y'' +p(t)y' +q(t)y = f(t).
A differential equation that cannot be written
in the form required by the definition of linear
DE is a nonlinear differential equation.
Classifying Differential Equations
Several differential equations can be classified
according with the properties we ’ve just introduced
| Equation |
Ord |
Lin/Nonlin |
Hom/Nonhom |
 |
linear
nonlinear
nonlinear
linear
nonlinear
linear |
non-hom
–
–
hom
–
nonhom |
Operator Notation
We introduce some simplifying notation, which
we will subsequently find very helpful.
We can write the linear algebraic equation (1)
simply as
L(x) = C,
where 
and the
operator L
is defined as
Similarly, we can write the linear differential
equation (2) simply as
L(y) = f(t),
where 
and the
operator
L is defined as
Linear Operator Property
We can easily confirm that the operators denoted
by L have the following properties, which
characterize linear operators:
Superposition Principle
Let 
and
be any solution of the homogeneous
linear equation
L(u) = 0:
Then their sum 
is also
a solution.
Furthermore a multiple 
is a solution
for any constant k.
Example 4
(a) The points (1, 3) and (2, 6) lie on the line
3x-y = 0. Check that their sum (1+2, 3+
6) = (3, 9) also lies on the same line.
(b) The points (2, 3, 5) and (-4,-5,-9) lie on
the plane x + y - z = 0. Check that their
sum (-2,-2,-4) also lies on the same plane.
(c) Two solutions of y'' - 4y = 0 are y = e2t
and y = e-2t. Check that y = e2t +e-2t is
also a solution.
Nonhomogeneous Principle
Let up be a particular solution to a linear
nonhomogeneous
equation
L(u) = C (algebraic)
or
L(u) = f(t) (differential).
Then
is also a solution, where 
is a solution to
the
associate homogeneous equation
L(u) = 0.
Furthermore, every solution of the nonhomogeneous
equation must be of the form .
Example 5 Consider the linear algebraic equation
x+y = 1.
The solutions of the associated homogeneous
equation x+y = 0 are Xh = c(1,-1), where c
is an arbitrary constant. A particular solution is
Xp = c(0, 1). Hence any solution is of the form
X = Xh+Xp = c(1,-1)+(0, 1) = (c, 1-c). Notice
that this is a line through the point (0, 1)
with direction given by the vector (1,-1).
Example 6 Consider the differential equation
1. Solve the associated homogeneous equation
y' - y = 0, to get
for any c ∈ IR.
2. Confirm, by differentiation and substitution
on (7), that
is a solution.
3. Form the sum and confirm that y = yh +
yp = cet - t - 1 is also a solution.
