f (x) = eax (P (x) cos bx +Q (x) sin bx)

,

( 3.29 )

n

m

Pn (x) and Qm (x)

are given binomial

of the

variable x

powered

correspondingly by n

and m . Let’s name

number

auxiliary

z =a+bi

y* = xr eax (P (x) cos bx +Q (x) sin bx)

,

l

l

where P (x) A xl

A xl 1

... A ,

Q (x) B

0

xl B xl 1

... B are

l

0

1

l

l

1

l

binomial

of

the

order

l max(n, m)

with

independent

coefficients A0 ,

A1 ,

..., Al ,

B0 , B1 , ..., Bl ; r is divisible by roots z i

Table 3.4

Structure of the particular

№

View of the right

Control number of the

side

right side

solution

( f (x) )

( z )

( y* )

1.1

e x

z – is not the root

Ae x

of the auxiliary equation

1.2

e x

z – the root of the

Ax r e x

auxiliary equation r

2.1

Pn (x)

z 0 – is not the root

P

(x) A xn

of the auxiliary equation

n

0

A xn 1 ... A

1

n

2.2

Pn (x)

z 0 – the root of the

x

P (x)

r ~

auxiliary equation

n

divisible by r

~

3.1

Pn (x)e

x

z – is not the root

x

of the auxiliary equation

Pn (x)e

223

3.2

Pn (x)e

x

z – is the root of

x

r ~

x

the auxiliary equation

Pn (x)e

divisible by r

A1 cos x B1 sin x

4.1

A cos bx + B sin bx

z i – is not the root

of the auxiliary equation

4.2

A cos bx + B sin bx

z i – root of the

x

r

( A1 cos x B1 sin

x)

auxiliary equation

divisible by r

x (Pn ( x) cos x

5.1

e x

cos xPn ( x)

z i – is not the

e

~

root of the auxiliary

~

equation

Q n ( x) sin x)

5.2

e x

cos xPn ( x)

z i – root of

x r e x (Pn ( x) cos x

~

the auxiliary equation

~

divisible by r

Q n ( x) sin x)

y a1 (x) y a2 (x) y f (x).

(3.30 )

Let y C1 y1 C2 y2 – be the general solution of

the homogeneous

y C1 (x) y1

C2 (x) y2 ,

(3.31)

where the functions C1 (x)

і

C2 (x)

depend on the system

ì

¢

¢

(x) y

= 0,

ïC (x) y +C

2

2

ï 1

1

(3.32)

í

¢(x) y ¢+C

¢

(x) y

¢

= f (x).

ïC

îï 1

1

2

2

Using Cramer’s formula can be evaluated

y1

y2

;

1

0

y2

;

2

y1

0

;

f (x)

f (x)

y1

y2

y2

y1

C1 (x)

1

;

C2

(x)

2

;

C1 (x) C1 (x)dx C3 ;

C2 (x) C2 (x)dx C4 .

224