Slide 1
Linear Differential Equations
Section 4.1
Preliminary Theory
Slide 2
What do we know about linear
differential equations?

They have a solution.

The solution is unique.

The solution is defined throughout an interval
I.
The above statements are true for an initial-value
problem. (i.e., we know y0, y0’ , y0’’ , etc.)
Slide 3
Boundary-Value Problems


This is a problem where the dependent variable or
its derivatives are specified at different points.
Example:
Solve a2 ( x) y  a1 ( x) y  a0 ( x) y  g ( x)
subject to:

y(a)  y0 , y(b)  y1
All bets are off! The statements on the previous
slide do not hold.
Slide 4
Examples

#2

#13
Slide 5
Two Types of Linear Equations
 Homogeneous
an ( x) y ( n )  an 1 ( x) y ( n 1)    a2 ( x) y  a1 ( x) y  a0 ( x) y  0
an ( x ) y

Nonhomogeneous
(n)
 an 1 ( x) y ( n 1)    a2 ( x) y  a1 ( x) y  a0 ( x) y  g ( x)
Later you will see that in order to solve a
nonhomogeneous differential equation, we must first
solve the associated homogeneous system.
Slide 6
Assumptions
We will always assume the following unless
stated otherwise:

The coefficients ai(x) are continuous.

g(x) is continuous.

a n(x) ≠ 0 for every x in the interval.
Slide 7
nth-Order Differential Operator
L  an ( x) D n  an 1 D n 1    a1 ( x) D  a0 ( x)
L is a linear operator.
i.e., L f ( x)   g ( x)   L f ( x)  Lg ( x)
Slide 8
Example
Express in terms of the D notation.
y  7 y  10 y  24e x
Slide 9
Superposition Principle
Any linear combination of solutions of a
homogeneous nth-order differential equation is
also a solution of the equation.
i.e., if y1, y2, …, yn are solutions then
y = c1y1 + c2y2 + … + cnyn
is also a solution where ci’s are arbitrary
constants.
Slide 10
Corollaries


A constant multiple of a solution is a
solution.
A homogeneous differential equation
always possesses the trivial solution
y = 0.
Slide 11
Linear Independence
A set of functions f1, f2, …, fn is linearly
dependent on an interval I if there exist
constants c1, c2, …, cn not all zero such that
c1f1 + c2f2 + … +cnfn = 0
for every x in I.
If a set of functions is not linearly dependent, it
is linearly independent.
Slide 12
Examples


Are f(t) = sin 2t and g(t) = (cos t)(sin t)
linearly independent?
Are f(t) = e3t and g(t) = t+1 linearly
independent?
A set of functions will be linearly dependent if
at least one of the functions can be written as
a linear combination of the others.
Slide 13
Wronskian and Linear
Independence
It turns out that a set of functions is
linearly independent on I if and only if
the Wronskian is not equal to 0 for every
x in the interval
Slide 14
Fundamental Set of Solutions
A set of n linearly independent solutions
of a homogeneous linear nth-order
differential equation is called a
fundamental set of solutions.
Theorem 4.4 states that a fundamental
set of solutions will always exist for a
homogeneous linear differential equation.
Slide 15
Example

#16
Slide 16
Theorem 4.5
Let y1, …., yn be a fundamental set of
solutions of a homogeneous nth-order
linear differential equation on I. Then the
general solution is
y = c1y1 + c2y2 + … + cnyn
where ci’s are arbitrary constants.
Slide 17
Example

#26
Slide 18
Theorem 4.6
Let yp be any particular solution of a
nonhomogeneous differential equation on I,
and let y1, …., yn be a fundamental set of
solutions of the associated homogeneous
equation. Then the general solution of the
equation on I is
y = c1y1 + c2y2 + … + cnyn + yp
i.e., y = complementary function + any particular solution
y = yc + yp
Slide 19
Example

#32

#36