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28) Á) y
00
+ y = 2 cos 4x + 3 sin 4x;
Â) y
000
− y
00
− 4y
0
+ 4y = (7 − 6x)e
x
;
×) y
00
+ y
0
=
e
x
2 + e
x
.
29) Á) y
00
+ 2y
0
+ y = 4x
2
;
Â) y
000
− y
00
− 9y
0
+ 9y = (12 − 16x)e
x
;
×) y
00
+ y =
1
cos x
.
30) Á) y
00
− 4y
0
+ 4y = e
2x
sin 6x;
Â) y
000
− 100y
0
= 20e
10x
;
×) y
00
+ 4y =
4
sin 2x
.
IX. îÁÊÔÉ ÏÂÝÅÅ ÒÅÛÅÎÉÅ ÓÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ
‘x =
dx
dt
; ‘y =
dy
dt
.
1.
‘x = −y
‘y = −4x.
2.
‘x = 2x + y
‘y = 3x + 4y.
3.
‘x = x − y
‘y = −4x + y.
4.
‘x = 3x + 2y
‘y = x + 2y.
5.
‘x = 3x − y
‘y = 4x −y.
6.
‘x = −3x + 2y
‘y = −2x + y.
7.
‘x = y
‘y = −x.
8.
‘x = x − 5y
‘y = x − y.
9.
‘x = x − 3y
‘y = 3x + y.
10.
‘x = x + y
‘y = −2x + 3y.
11.
‘x = 5x + 4y
‘y = 4x + 5y.
12.
‘x = y
‘y = x.
13.
‘x = 3x − 2y
‘y = 2x −y.
14.
‘x = 2x + y
‘y = 3x + 4y.
15.
‘x = 2x + 3y
‘y = 6x −y.
16.
‘x = 3x + 5y
‘y = −2x −8y.
17.
‘x = x + y
‘y = 2x −y.
18.
‘x = 2x + 3y
‘y = x −3y.
19.
‘x = 3x − 2y
‘y = 5x + 6y.
20.
‘y + 2x − 3y = 0
‘x + x + 4y = 0.
21.
‘x + ‘y + 2x −3y = 0
2 ‘x −3 ‘y + x − 2y = 0.
22.
‘x + x − 8y = 1
‘y − x − y = 0.
23.
‘x + x + 5y = 0
‘y − x − y = 0.
24.
‘x = 2x + y
‘y = 4y − x.
25.
‘x = 3x − y
‘y = 4x − y.
26.
‘x − 5x − 3y = 0
‘y + 3x + y = 0.
27.
‘x = 4x − y
‘y = x + 2y.
28.
‘x = 3x + 2y
‘y = x + 2y.
29.
‘x = 2x − 4y
‘y = 2x −2y.
30.
‘x = 5x − 3y
‘y = x + y.
57
28) Á) y 00 + y = 2 cos 4x + 3 sin 4x;
Â) y 000 − y 00 − 4y 0 + 4y = (7 − 6x)ex;
ex
×) y 00 + y 0 = .
2 + ex
29) Á) y 00 + 2y 0 + y = 4x2;
Â) y 000 − y 00 − 9y 0 + 9y = (12 − 16x)ex;
1
×) y 00 + y = .
cos x
30) Á) y 00 − 4y 0 + 4y = e2x sin 6x;
Â) y 000 − 100y 0 = 20e10x;
4
×) y 00 + 4y = .
sin 2x
dx dy
IX. îÁÊÔÉ ÏÂÝÅÅ ÒÅÛÅÎÉÅ ÓÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ x‘ = ; y‘ = .
dt dt
x‘ = −y x‘ = 2x + y x‘ = x − y
1. 2. 3.
y‘ = −4x. y‘ = 3x + 4y. y‘ = −4x + y.
x‘ = 3x + 2y x‘ = 3x − y x‘ = −3x + 2y
4. 5. 6.
y‘ = x + 2y. y‘ = 4x − y. y‘ = −2x + y.
x‘ = y x‘ = x − 5y x‘ = x − 3y
7. 8. 9.
y‘ = −x. y‘ = x − y. y‘ = 3x + y.
x‘ = x + y x‘ = 5x + 4y x‘ = y
10. 11. 12.
y‘ = −2x + 3y. y‘ = 4x + 5y. y‘ = x.
x‘ = 3x − 2y x‘ = 2x + y x‘ = 2x + 3y
13. 14. 15.
y‘ = 2x − y. y‘ = 3x + 4y. y‘ = 6x − y.
x‘ = 3x + 5y x‘ = x + y x‘ = 2x + 3y
16. 17. 18.
y‘ = −2x − 8y. y‘ = 2x − y. y‘ = x − 3y.
x‘ = 3x − 2y y‘ + 2x − 3y = 0 x‘ + y‘ + 2x − 3y = 0
19. 20. 21.
y‘ = 5x + 6y. x‘ + x + 4y = 0. 2x‘ − 3y‘ + x − 2y = 0.
x‘ + x − 8y = 1 x‘ + x + 5y = 0 x‘ = 2x + y
22. 23. 24.
y‘ − x − y = 0. y‘ − x − y = 0. y‘ = 4y − x.
x‘ = 3x − y x‘ − 5x − 3y = 0 x‘ = 4x − y
25. 26. 27.
y‘ = 4x − y. y‘ + 3x + y = 0. y‘ = x + 2y.
x‘ = 3x + 2y x‘ = 2x − 4y x‘ = 5x − 3y
28. 29. 30.
y‘ = x + 2y. y‘ = 2x − 2y. y‘ = x + y.
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