Операционное исчисление. - 15 стр.

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МГТУ ГА | Москва

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§4. úÁÄÁÞÁ ëÏÛÉ ÄÌÑ ÌÉÎÅÊÎÙÈ ÕÒÁ×ÎÅÎÉÊ 15
ïÔÓÀÄÁ
X(p) =
2p
(p
2
+ 1)
2
−
1
p
2
+ 1
.
îÁÈÏÄÉÍ ÏÒÉÇÉÎÁÌ ÄÌÑ X(p) :
1
p
2
+1
← sin t,
2p
(p
2
+1)
2
← t sin t.
úÎÁÞÉÔ X(p) →t sin t − sin t = (t − 1) sin t É ÐÏÌÕÞÁÅÍ ÒÅÛÅÎÉÅ ÉÓÈÏÄÎÏÇÏ
ÕÒÁ×ÎÅÎÉÑ
x(t) = (t − 1) sin t.
úÁÄÁÞÉ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ÒÅÛÅÎÉÑ
òÅÛÉÔØ ÓÌÅÄÕÀÝÉÅ ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ ÐÒÉ ÚÁÄÁÎÎÙÈ ÎÁÞÁÌØÎÙÈ
ÕÓÌÏ×ÉÑÈ:
91) x
00
+ 3x
0
= e
t
, x(0) = 0, x
0
(0) = −1.
92) x
00
− 2x
0
= e
2t
, x(0) = x
0
(0) = 0.
93) x
00
+ 2x
0
− 3x = e
−t
, x(0) = 0, x
0
(0) = 1.
94) x
000
+ x
0
= 1, x(0) = x
0
(0) = x
00
(0) = 0.
95) x
00
+ 2x
0
= t sin t, x(0) = x
0
(0) = 0.
96) x
00
+ 2x
0
+ x = sin t, x(0) = 0, x
0
(0) = −1.
97) x
000
− x
00
= sin t, x(0) = x
0
(0) = x
00
(0) = 0.
98) x
00
− 2x
0
+ x = e
t
, x(0) = 0, x
0
(0) = 1.
99) x
000
+ 2x
00
+ 5x
0
= 0, x(0) = −1, x
0
(0) = 2.
100) x
00
− 2x
0
+ 2x = 1, x(0) = x
0
(0) = 0.
101) x
00
+ x
0
= cos t, x(0) = 2, x
0
(0) = 0.
102) x
00
+ 2x
0
+ x = t
2
, x(0) = 1, x
0
(0) = 0.
103) x
000
+ x
00
= sin t, x(0) = x
0
(0) = 1, x
00
(0) = 0.
104) x
00
+ x = cos t, x(0) = −1, x
0
(0) = 1.
105) x
000
+ x
00
= t, x(0) = −3, x
0
(0) = 1, x
00
(0) = 0.
106) x
00
+ 2x
0
+ 5x = 3, x(0) = 1, x
0
(0) = 0.
107) x
IV
− x
00
= cos t, x(0) = 0, x
0
(0) = −1, x
00
(0) = x
000
(0) = 0.
108) x
00
+ x = 1, x(0) = −1, x
0
(0) = 0.
109) x
00
+ 2x
0
+ 2x = 1, x(0) = x
0
(0) = 0.
110) x
00
+ 4x = t, x(0) = 1, x
0
(0) = 0.
111) x
00
− 2x
0
+ 5x = 1 − t, x(0) = x
0
(0) = 0.
112) x
000
+ x = 0, x(0) = 0, x
0
(0) = −1, x
00
(0) = 2.
113) x
000
+ x
00
= cos t, x(0) = −2, x
0
(0) = x
00
(0) = 0.
114) x
000
+ x
0
= e
t
, x(0) = 0, x
0
(0) = 2, x
00
(0) = 0.
115) x
IV
− x
00
= 1, x(0) = x
0
(0) = x
00
(0) = x
000
(0) = 0.
§4. úÁÄÁÞÁ ëÏÛÉ ÄÌÑ ÌÉÎÅÊÎÙÈ ÕÒÁ×ÎÅÎÉÊ                                 15

ïÔÓÀÄÁ
                                     2p       1
                        X(p) =            −       .
                                 (p2 + 1)2 p2 + 1
îÁÈÏÄÉÍ ÏÒÉÇÉÎÁÌ ÄÌÑ X(p) :
                                 1
                              p2 +1     ← sin t,
                                2p
                             (p2 +1)2   ← t sin t.
úÎÁÞÉÔ X(p) → t sin t − sin t = (t − 1) sin t É ÐÏÌÕÞÁÅÍ ÒÅÛÅÎÉÅ ÉÓÈÏÄÎÏÇÏ
ÕÒÁ×ÎÅÎÉÑ
                              x(t) = (t − 1) sin t.

úÁÄÁÞÉ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ÒÅÛÅÎÉÑ

òÅÛÉÔØ ÓÌÅÄÕÀÝÉÅ ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ ÐÒÉ ÚÁÄÁÎÎÙÈ ÎÁÞÁÌØÎÙÈ
ÕÓÌÏ×ÉÑÈ:
   91) x00 + 3x0 = et , x(0) = 0, x0 (0) = −1.
   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 sin t, x(0) = x0(0) = 0.
   96) x00 + 2x0 + x = sin t, x(0) = 0, x0 (0) = −1.
   97) x000 − x00 = sin t, 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 = cos t, x(0) = 2, x0(0) = 0.
   102) x00 + 2x0 + x = t2 , x(0) = 1, x0(0) = 0.
   103) x000 + x00 = sin t, x(0) = x0 (0) = 1, x00(0) = 0.
   104) x00 + x = cos t, 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) xIV − x00 = cos t, 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 = cos t, x(0) = −2, x0 (0) = x00(0) = 0.
   114) x000 + x0 = et , x(0) = 0, x0(0) = 2, x00 (0) = 0.
   115) xIV − x00 = 1, x(0) = x0(0) = x00 (0) = x000(0) = 0.

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