Slide 1
Homogeneous Linear Equations
with Constant Coefficients
Differential Equations
Section 4.3
Slide 2
Homogeneous Linear Equations
with Constant Coefficients
A differential equation of the form
ay   by   cy  0
can always be solved in terms of elementary
functions of calculus.
If the coefficients are not constant they are
usually much more difficult.
Possibilities for y: et, e-t, cet, ce-t, A linear
combination thereof
Slide 3
Example
Can you think of a solution of the differential
equation given below?
y   y  0
How many different solutions can you come up
with just using trial and error?
Slide 4
General Case
What about the equation
ay   by   cy  0 ?
We will form the auxiliary equation
am2 + bm + c = 0.
There are three possibilities for the roots m1 and m2 of this
equation:
 m1 and m2 are real and distinct
 m1 and m2 are real and equal
 m1 and m2 are conjugate complex numbers
Slide 5
Case I: Distinct Real Roots
In this case we find two solutions,
y1  e m1x
and
y2  e m2 x
These functions are linearly independent on (-∞,∞) and hence form a
fundamental set of solutions.
y  c1e  c2e
Any equation of the form
where m1 and m2 are distinct real solutions of the auxiliary equation
will be a solution of the given differential equation.
m1t
Example:
m2t
4 y   9 y  0
Slide 6
Case II: Repeated Real Roots
In this case we find two solutions,
y1  e m1x
and
y2  xem1x
These functions are linearly independent on (-∞,∞) and hence form a
fundamental set of solutions.
y  c1e 1  c2 xe 1
Any equation of the form
where m1 = m2 are repeated real solutions of the auxiliary equation will
be a solution of the given differential equation.
mt
Example:
y  10 y  25 y  0
mt
Slide 7
Case III: Conjugate Complex Roots
In this case we find two solutions, y  e m1x and y  e m2 x
1
2
where m     i and m    i .
2
1
Using Euler's formula we can rewrite these solutions as
y  2e x cos  x and y  2e x sin  x .
1
2
These functions are linearly independent on (-∞,∞) and hence form a
fundamental set of solutions, so the general solution will be
y  c1e x cos  x  c2e x sin  x
Example:
y  2 y  2 y  0,
y(0)  1,
y (0)  1
Slide 8
Higher-Order Equations



Form the auxiliary equation and find all the roots.
You may have to use synthetic division.
The methods of writing the solutions of the
differential equations on the previous slide are
just extended. Refer to pages 145 and 146 in
your text.
Example:
y  2 y  5 y  6 y  0,
y (0)  y(0)  0,
y(0)  1