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y
00
− 9y
0
= e
x
(3x + 7)
y
00
− 9y = sin 3x − 2 cos 3x
y
00
+ 4y =
1
sin 2x
˙x = 6x + 2y
˙y = −2x + 2y
a x
y =
1
2
(x
2
+ 1)
1 + (y
0
)
2
= 2yy
00
t
ds
dt
= s +
√
st
xy(1 + x
2
)y
0
= 1 + y
2
y
0
+ 2y = 4x
y
x
dx + (y
3
+ ln x)dy = 0
y
0
+
2y
x
= 3x
2
y
4
3
(y
2
− x
2
)dx + 2xydy = 0
yy
00
− (y
0
)
2
= y
4
.
y
00
+ 2x(y
0
)
2
= 0.
y
00
+ 4y = sin 2x + 6 cos 2x
y
00
− 10y
0
+ 25y = (x − 2)e
−5x
y
00
+ y =
1
cos
3
x
y
00
+ 8y
0
+ 25y = cos 2x
˙x = x − 3y
˙y = 4x − 10y
a ·l l
y = e
−2x
y
00
− 4y = 0
10. y 00 − 9y 0 = ex (3x + 7).
11. y 00 − 9y = sin 3x − 2 cos 3x.
1
12. y 00 + 4y = .
sin 2x
ẋ = 6x + 2y
13.
ẏ = −2x + 2y
14. Íàéòè êðèâóþ, ïðîõîäÿùóþ ÷åðåç íà÷àëî êîîðäèíàò, åñëè äëÿ
ëþáîãî îòðåçêà [a, x] ïëîùàäü êðèâîëèíåéíîé òðàïåöèè, îãðàíè÷åííîé
äóãîé ýòîé êðèâîé, ðàâíà êóáó îðäèíàòû êîíöåâîé òî÷êè äóãè.
1
15. Ïîêàçàòü, ÷òî ôóíêöèÿ y = (x2 + 1) ÿâëÿåòñÿ ðåøåíèåì äèôôå-
2
ðåíöèàëüíîãî óðàâíåíèÿ 1 + (y 0 )2 = 2yy 00 .
Âàðèàíò 30.
ds √
1. t = s + st.
dt
2. xy(1 + x2 )y 0 = 1 + y 2 .
3. y 0 + 2y = 4x.
y
4. dx + (y 3 + ln x)dy = 0.
x
2y 4
5. y 0 + = 3x2 y 3 .
x
6. (y 2 − x2 )dx + 2xydy = 0.
7. yy 00 − (y 0 )2 = y 4 .
8. y 00 + 2x(y 0 )2 = 0.
9. y 00 + 4y = sin 2x + 6 cos 2x.
10. y 00 − 10y 0 + 25y = (x − 2)e−5x .
1
11. y 00 + y = .
cos3 x
12.
y + 8y + 25y = cos 2x.
00 0
ẋ = x − 3y
13.
ẏ = 4x − 10y
14. Íàéòè êðèâóþ, ïðîõîäÿùóþ ÷åðåç òî÷êó À(0, à), åñëè ïëîùàäü
OAMN ðàâíà a · l, ãäå MNïðîèçâîëüíàÿ îðäèíàòà êðèâîé, läëèíà äóãè
AM.
15. Ïîêàçàòü, ÷òî ôóíêöèÿ y = e−2x ÿâëÿåòñÿ ðåøåíèåì äèôôåðåí-
öèàëüíîãî óðàâíåíèÿ y 00 − 4y = 0.
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