second order linear differential equation with constant coefficients

second order linear differential equation with constant coefficients

Consider the

second order

homogeneous linear differential equation

x

′′

+

b

⁢

x

′

+

c

⁢

x

=

0

,

(1)

where

b

and

c

are real constants.

The explicit solution is easily found using the

characteristic equation

method. This method, introduced by Euler, consists in seeking solutions of the form

x

⁢

(

t

)

=

e

r

⁢

t

for (

1

). Assuming a solution of this form, and substituting it into (

1

) gives

r

2

⁢

e

r

⁢

t

+

b

⁢

r

⁢

e

r

⁢

t

+

c

⁢

e

r

⁢

t

=

0

.

Thus

r

2

+

b

⁢

r

+

c

=

0

(2)

which is called the

characteristic equation

of (

1

). Depending on the nature of the roots (

http://planetmath.org/Equation

)

r

1

and

r

2

of (

2

), there are three cases.

•

If the roots are real and distinct, then two

linearly independent

solutions of (

1

) are

x

1

⁢

(

t

)

=

e

r

1

⁢

t

,

x

2

⁢

(

t

)

=

e

r

2

⁢

t

.

•

If the roots are real and equal, then two linearly independent solutions of (

1

) are

x

1

⁢

(

t

)

=

e

r

1

⁢

t

,

x

2

⁢

(

t

)

=

t

⁢

e

r

1

⁢

t

.

•

If the roots are

complex conjugates

of the form

r

1

,

2

=

α

±

i

⁢

β

, then two linearly independent solutions of (

1

) are

x

1

⁢

(

t

)

=

e

α

⁢

t

⁢

cos

⁡

β

⁢

t

,

x

2

⁢

(

t

)

=

e

α

⁢

t

⁢

sin

⁡

β

⁢

t

.

The

general solution

to (

1

) is then constructed from these linearly independent solutions, as

ϕ

⁢

(

t

)

=

C

1

⁢

x

1

⁢

(

t

)

+

C

2

⁢

x

2

⁢

(

t

)

.

(3)

Characterizing the behavior of (

3

) can be accomplished by studying the two-dimensional

linear system

obtained from (

1

) by defining

y

=

x

′

:

x

′

=

y

(4)

y

′

=

-

b

⁢

y

-

c

⁢

x

.

(5)

Remark that the roots of (

2

) are the

eigenvalues

of the Jacobian matrix of (

5

). This generalizes to the characteristic equation of a

differential equation

of order

n

and the

n

-dimensional system associated to it.

Also note that the only equilibrium of (

5

) is the origin

(

0

,

0

)

. Suppose that

c

≠

0

. Then

(

0

,

0

)

is called a

r

o

m

a

n

]

e

n

u

m

e

r

a

t

e

source

i

f

f

b

<

0

a

n

d

c

>

0

,

spiral source

i

f

f

i

t

i

s

a

s

o

u

r

c

e

a

n

d

b

2

-

4

c

<

0

,

sink

i

f

f

b

>

0

a

n

d

c

>

0

,

spiral sink

i

f

f

i

t

i

s

a

s

i

n

k

a

n

d

b

2

-

4

c

<

0

,

i

f

f

c

<

0

,

center

i

f

f

b

=

0

a

n

d

c

>

0

.

Title

second order linear differential equation with constant coefficients

Canonical name

SecondOrderLinearDifferentialEquationWithConstantCoefficients

Date of creation

2013-03-22 13:24:49

Last modified on

2013-03-22 13:24:49

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Topic

Classification

msc 34A30

Classification

msc 34-01

Classification

msc 34C05

Related topic

GeneralSolutionOfLinearDifferentialEquation

Related topic

TelegraphEquation

Defines

characteristic equation

Defines

source

Defines

sink

Defines

center

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