Question

discuss the general method of solving a second order homogeneous differential equation with constant coefficients​ please solve

Answer 1

Answer:

The general solution of the homogeneous differential equation depends on the roots of the characteristic quadratic equation. There are the following options:

Discriminant of the characteristic quadratic equation

D

>

0.

Then the roots of the characteristic equations

k

1

and

k

2

are real and distinct. In this case the general solution is given by the following function

y

(

x

)

=

C

1

e

k

1

x

+

C

2

e

k

2

x

,

where

C

1

and

C

2

are arbitrary real numbers.

Discriminant of the characteristic quadratic equation

D

=

0.

Then the roots are real and equal. It is said in this case that there exists one repeated root

k

1

of order 2. The general solution of the differential equation has the form:

y

(

x

)

=

(

C

1

x

+

C

2

)

e

k

1

x

.