Операционное исчисление. Учебное пособие

x3. • 宦¤¥-¨¥ ®à¨£¨- «

¯® ¨§®¡à ¦¥-¨î

11

69)

F (p) =

e 2p

:

p 3p1

70)

F (p) =

e

:

p + 3

71)

F (p) =

1

:

p2 + 4p + 5

72)

F (p) =

1

:

p2 + 4p + 3

73)

F (p) =

1

:

(p2 + 1)2

74)

F (p) =

p

:

(p2 + 1)2

75)

F (p) =

1

:

p + 2p2 + p3

76)

F (p) =

1

:

7

3

p + p2

2

77)

F (p) =

2p

+ p

+ 2p + 2

:

p5 + 2p4 + 2p3

78)

F (p) =

1

:

p2(p2 + 1)

79)

F (p) =

p + 2

:

(p + 1)(p 2)(p2 + 4)

80)

F (p) =

1

:

p4 + 2p3 + 3p2 + 2p + 1

81)

F (p) =

p2 + 2p 1

:

p3 + 3p2 + 3p + 1

82)

F (p) =

p

:

p3 + 1

83)

F (p) =

2p + 3

:

p3 + 4p2 + 5p

84)

F (p) =

e 3p

:

(p + 1)2

85)

F (p) =

e p

:

p(p 1)

86)

F (p) =

1

e 2p + 2e 3p + 3e 4p :

p2 + 1

87)

F (p) =

e p

+

pe 2p

:

p2 1

e

pp2 4

88)

F (p) =

2

:

p(p + 1)(p2 + 4)

12

x4. ‡ ¤ ç Š®è¨ ¤«ï «¨-¥©-ëå ãà ¢-¥-¨©

e p

2e 2p

6e 3p

89)

F (p) =

+

+

:

p2

p3

p4

90)

F (p) =

e 3p

:

p(p2 + 1)

(1)

a0

dnx(t)

+ a1

dn 1x(t)

+ : : : + an 1

dx(t)

+ anx(t) = f (t):

dtn

dtn 1

dt

(2)

x(0) = x0; x0(0) = x0

; : : : ; x(n 1)(0) = x(n 1):

0

0

(4)

X(p) =

F (p)+ n 1(p)

;

'n(p)

x4. ‡ ¤ ç Š®è¨ ¤«ï «¨-¥©-ëå ãà ¢-¥-¨©

13

a0

d2x(t)

+ a1

dx(t)

+ a2x(t) = f (t)

dt2

dt

14

x4. ‡ ¤ ç Š®è¨ ¤«ï «¨-¥©-ëå ãà ¢-¥-¨©

•¥è¥-¨¥: •ãáâì x(t)! X(p): ’ ª ª ª f (t) = t; â® t !

1

:

p2

„ «¥¥ ¯®«ãç ¥¬

dxdt

! pX(p) x(0) = pX(p);

d2x

! p2X(p) px(0) x0(0) = p2X(p):

dt2

¬¥-â à-ë¥ ¤à®¡¨

1

=

A

+

B

+

C

+

D

;

p2(p2 + 3p + 2)

p

p2

p + 1

p + 2

-¨©

8

A

C + D

= 0;

3A + B ++2C + D = 0;

A =

5

3; B =

1;

>

A + 3B = 0;

=

C =

2

2

>

)

2

; D =

1:

<

2B

=

1:

>

>

’®£¤ :

3

1

1

1

5

1

1

X(p) =

+

+

2

p

2

p2

2

p + 1

p + 2

¨ ¯® ä®à¬ã« ¬ ??, ?? ¨ ?? â ¡«¨æë ¨§®¡à ¦¥-¨© -

á. ?? - 室¨¬ à¥è¥-¨¥

3

1

5

x(t) =

+

t +

e t e 2t:

2

2

2

•à¨¬¥à 2. •¥è¨âì § ¤ çã Š®è¨

x00 + x = 2cost;

x(0) = 0;

x0(0) = 1:

•¥è¥-¨¥:

! X(p);

x(t)

x0(t)

! pX(p) x(0) = pX(p);

x00(t)

! p2X(p) px(0) x0(0) = p2X(p) + 1;

cost !

p

p2X(p) + 1 + X(p) =

2p

;

:

p2 + 1

p2 + 1

x4. ‡ ¤ ç Š®è¨ ¤«ï «¨-¥©-ëå ãà ¢-¥-¨©

15

Žâáî¤

2p

1

X(p) =

:

(p2 + 1)2

p2 + 1

• 室¨¬ ®à¨£¨- « ¤«ï X(p) :

1

sint;

p2+1

2p

t sint:

(p2+1)2

ãá«®¢¨ïå:

= et; x(0) = 0;

x0(0) =

1:

91) x00

+ 3x0

92) x00 2x0

= e2t; x(0) = x0(0) = 0:

93) x00

+ 2x0

3x = e t;

x(0)

= 0;

x0(0) = 1:

94) x000

+ x0 = 1; x(0) = x0(0)

= x00(0) = 0:

95) x00

+ 2x0

= t sint;

x(0) = x0(0)

= 0:

96) x00

+ 2x0

+ x = sint;

x(0) = 0;

x0(0) = 1:

97) x000

x00

= sint;

x(0) = x0(0) = x00(0) = 0:

98) x00 2x0 + x = et;

x(0) = 0; x0(0) = 1:

99) x000 + 2x00

+ 5x0

= 0; x(0) = 1; x0(0) = 2:

100) x00

2x0

+ 2x = 1; x(0) = x0(0) = 0:

101) x00

+ x0 = cost;

x(0) = 2;

x0(0) = 0:

102) x00

+ 2x0

+ x = t2; x(0) = 1; x0(0) = 0:

103) x000

+ x00

= sint;

x(0) = x0(0) = 1; x00(0) = 0:

104) x00

+ x = cost;

x(0) = 1; x0(0) = 1:

105) x000

+ x00

= t;

x(0) = 3;

x0(0) = 1; x00(0) = 0:

106) x00

+ 2x0

+ 5x = 3; x(0) = 1; x0(0) = 0:

107) xI V x00

= cost;

x(0) = 0;

x0(0) = 1; x00(0) = x000(0) = 0:

108) x00

+ x = 1;

x(0) = 1;

x0(0) = 0:

109) x00

+ 2x0

+ 2x = 1; x(0) = x0(0) = 0:

110) x00

+ 4x = t;

x(0) = 1;

x0(0) = 0:

111) x00

2x0

+ 5x = 1 t;

x(0) = x0(0) = 0:

112) x000

+ x = 0;

x(0) = 0;

x0(0) = 1; x00(0) = 2:

113) x000

+ x00

= cost;

x(0) = 2; x0(0) = x00(0) = 0:

114) x000

+ x0 = et;

x(0) = 0;

x0(0) = 2; x00(0) = 0:

115) xI V x00

= 1; x(0) = x0(0) = x00(0) = x000(0) = 0:

16

x4.

‡ ¤ ç

Š®è¨ ¤«ï «¨-¥©-ëå ãà ¢-¥-¨©

116) x00

+ x0 = cost;

x(0) = 2;

x0(0) = 0:

117) x00

x0 = tet; x(0) = x0(0) = 0:

2; x00(0) = 0:

118) x000

+ x0 = cost;

x(0) = 0;

x0(0) =

119) x00

+ 2x0 + x = t;

x(0) = x0(0) = 0:

120) x00

x0 + x = e t; x(0) = 0; x0(0) = 1:

121) x00

x = sint;

x(0) = 1;

x0(0) = 0:

122) x000

+ x = et;

x(0) = 0;

x0(0) = 2;

x00(0) = 0:

123) x00

+ x = 2sint;

x(0) = 1;

x0(0) = 1:

124) x00

2x0 + x = t sint;

x(0) = x0(0) = 0:

125) x00

+ 2x0 + x = 2cos2 t;

x(0) = x0(0) = 0:

126) x00

+ 4x = 2cost cos3t;

x(0) = x0(0) = 0:

127) x00

+ x = tet + 4sint;

x(0) = x0(0) = 0:

128) x00

x0 = tet;

x(0) = 1;

x0(0) = 0:

129) x00

+ x0 = 4sin2 t;

x(0) = 0;

x0(0) = 1:

130) x000

2x00 + x0 = 4;

x(0) = 1; x0(0) = 2; x00(0) = 2:

131) x00

3x0 + 2x = et;

x(0) = x0(0) = 0:

132) x00

x0 = t2;

x(0) = 0;

x0(0) = 1:

133) x000

+ x =

1

t2et;

x(0) = x0(0) = x00(0) = 0:

2

134) x00

x(0) = x0(0) = 0:

+ x = t cos2t;

135) x00

+ n2x = a sin(nt + );

x(0) = x0(0) = 0:

136) x000

+ 6x00

+ 11x0 + 6x = 1 + t + t2; x(0) = x0(0) = x00(0) = 0:

137) xI V + 2x00

+ x = t sint;

x(0) = x0(0) = x00(0) = x000(0) = 0:

138) x00

2 x0 + ( 2 + 2)x = 0;

x(0) = 0; x0(0) = 1:

139) x00

+ 4x = sint;

x(0) = x0(0) = 0:

140) x000 + x0 = e2t; x(0) = x0(0) = x00(0) = 0:

141) xI V + x000

= cost;

x(0) = x0(0) = x00(0) = 0; x000(0) = :

142) x00

4x = sin

3

sin

1

x(0) = 1;

x0(0) = 0:

t

t;

2

2

143) xI V 5x00 +10x0 6x = 0; x(0) = 1; x0(0) = 0; x00(0) = 6; x000(0) = 14:

144) x00

+ x0 + x = tet;

x(0) = x0(0) = 0:

145) x00

+ x = t cost;

x(0) = x0(0) = 0:

146) x000

+ 3x00 4x = 0;

x(0) = x0(0) = 0; x00(0) = 2:

147) x000

+ 3x00 + 3x0 + x = 1; x(0) = x0(0) = x00(0) = 0:

148) x000

+ x = 1;

x(0) = x0(0) = x00(0) = 0:

x5. ‡ ¤ ç Š®è¨ ¤«ï «¨-¥©-ëå á¨á⥬

17

n

aik

d2x

dx

= fi(t) (i = 1; 2; : : : ; n);

X

k=1

n

X

aik p2

+ bik p + cik = Fi(p)+

k=1

n

X

+

[(aik p + bik) k + aik k ] (i = 1; 2; : : : ; n):

k=1

•¥è ï íâã á¨á⥬㠪 ª «¨-¥©-ãî

«£¥¡à ¨ç¥áªãî á¨á⥬ã ãà ¢-¥-¨© ®â-®-

•à¨¬¥à 1. •¥è¨âì á¨á⥬ã ãà ¢-¥-¨©

8 y00

= x

y;

<

x00

= 3(y

x + z);

¯à¨ - ç «ì-ëå ãá«®¢¨ïå

: z00

=

z

x(0)

=

0;

x0(0)

=

0;

y(0)

=

0;

y0(0)

=

1;

z(0)

=

1;

z0(0)

=

0:

X(p) =

3(2 p2 1)

;

p (p +4)

3(p 1)

1

Y (p) =

p2(p2+1)(p2+4)

p2+1

;

Z(p) =

p

:

2

p +1

• ©¤•¥¬ ®à¨£¨- «ë íâ¨å ¨§®¡à ¦¥-¨©.

X(p) =

3(p 1)

:

p2(p2 + 4)

X(p)

=

A +

B2

+

C

:

2

3(p 1)

p

p

p +4

=

Ap(p2 + 4)+ B(p2 + 4) + (Cp + D)p2:

p3

=

A + C;

0

p2

0

= B + D;

=

A = 43; B = 43;

p

3

= 4A;

)

D =

3

; C =

3

:

p0

3

=

4B:

4

4

3

1

3

1

3

p

3

1

X(p) =

+

:

4

p

4

p2

4

p2 + 4

4

p2 + 4

• å®¤ï ®à¨£¨- « ¤«ï ª ¦¤®£® á« £ ¥¬®£®, ¯®«ã稬

3

3

3

3

x(t) =

t

cos2t +

sin2t:

4

4

4

8

„ «¥¥ - ©¤•¥¬ ®à¨£¨- « ¤«ï Y (p):

Y (p) =

3(p 1)

1

;

1

sint:

p2(p2 + 1)(p2 + 4)

p2 + 1

p2 + 1

!

Ž¡®§- 稬

Y1(p) =

3(p 1)

:

p2(p2 + 1)(p2 + 4)