Сборник задач по высшей математике 2 том
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Pew;um'b CUCmeM'bt ypa6Hetm11:
2.8.13. {:': t' |
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dY |
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2.8.14. |
{ dx = (z _ y)2' |
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(z _ y)2' |
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y2 |
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2.8.15. |
{ dt |
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{:~ = z~l,
2.8.16.dz = _1_ y(O) = -1, z(O) = 1.
dx y - x'
yl = -z,
2.8.17. PemHTh cHcTeMY { Z' = Z2 |
r.n;e y = y(t), |
z = z(t). |
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Y' |
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_Z'. TIO.n;CTaBHB z' H3 |
TIpo.n;H<p<pepeHlJ;HpyeM nepBoe ypaBHeHHe: y" = |
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BTOporo ypaBHeHHfI CHCTeMhI, nOJIY'IaeM y" = - t2 . TIOCKOJIhKY H3 nep- |
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BorO ypaBHeHHfI (yl)2 = Z2, npHxo.n;HM K ypaBHeHHIO OTHOCHTeJIhHO O,ll;Hoit |
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T'(yl)2
<PYHKIJ;HH: y" = HJIH yy" + (yl)2 = O. 3TO ypaBHeHHe paBHOCHJIhHO ypaBHeHHIO (yy')' = 0, oTKy.n;a yy' = Cl. Pa3.n;eJIfifi nepeMeHHhIe, nOJIy'IHM
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y2 |
+ C2, T. e. y = ±J2(C l t + C2). <l>YHKIJ;HIO z |
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y dy = Cl dt, oTKy.n;a "'2= Clt |
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HaxO.n;HM H3 nepBoro ypaBHeHHfI HCXO.n;HOit CHCTeMhI: |
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M OKOH'IaTeJIhHO, |
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y = ±J2(C l t + C2), |
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±C l |
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Z= |
2..jCl t + C2 |
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2.8.18. |
X' |
= 2x + 3y, |
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PemHTh CHCTeMY { I |
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y = 6x - y. |
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a ,1l;aHHYIO CHCTeMY pemHM MaTpH'IHhIMcnoco6oM, HCnOJIh3Yfl c06cTBeHHhIe |
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'IHCJIaH c06cTBeHHhIe BeKTophI MaTpHIJ;hI npaBoit 'IaCTHCHCTeMhI. 0603Ha'IHM 'Iepe3
A= |
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3) |
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-1 |
MaTpHIJ;y CHCTeMhI.
CocTaBHM XapaKTepHCTH'IeCKOeypaBHeHHe det(A - kE) = 0, T. e.
2 - k |
3 1 |
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- l - k =0. |
TIPHXO.n;HM K ypaBHeHHIO k2 - |
k - |
20 = 0 C KOpHflMH kl = -4, k2 = 5. |
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HaxO,ll;HM C06CTBeHHbIe BeKTOpbI. IIpH k = -4 HMeeM: 6C1 + 3C2 = 0,
C1 =1, C2 = -2 H co6cTBeHHbIft BeKTop HMeeT BH,ll; (-~~J:
IIpH k = 5 HMeeM: -3C1 +3C2 = 0, C1 = 1, C2 = 1, c06cTBeHHbIft BeKTOp
HMeeT BH,ll; (g:).
CocTaBJUIeM o6rn;ee peweHHe CHCTeMbI
(X) |
(C) |
k1X |
+ |
(C ) |
k2X |
= |
( Ce |
k1X+ Ce |
k2X |
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y |
= -2~1 e |
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C: e |
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-2C1e |
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+ C2 e |
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1k1X |
2 k2X |
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T.e. |
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X = C1e-4x + C2 e5x , |
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y = -2C1e-4x + C2 e5x • |
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X' = x+y, |
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2.8.19. PewHTb cHcTeMY { y' = -x +y - z, |
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z' = 3y + z. |
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a MaTpHu;a CHCTeMbI HMeeT BH,ll; |
i ~1) |
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A= (~1 |
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OTCIO,ll;a XapaKTepHCTH'IeCKOeypaBHeHHe |
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l-k |
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-1 |
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= 0 |
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l-k |
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C XapaKTepHCTH'IeCKHMH'IHCJIaMHk1 = 1, k2 = 1 +2i, k3 = 1 - |
2i. |
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Co6cTBeHHbIft BeKTOp, OTBe'laIOIII;Hftco6cTBeHHoMY 'IHCJIYk1 |
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= 1, nOJIy- |
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qaeM H3 CHCTeMbI
T.e.
npe,ll;CTaBJIjleT co60ft HOpMHpoBaHHbIft (e,n;HHH'IHbIft) BeKTop OTBe'laIOrn;Hft Co6cTBeHHoMY 'IHCJIYk1 = 1 (XOTjI nepeXO,ll;HTb K e,ll;HHH'IHbIMBeKTopaM He06j13aTeJIbHO) •
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Co6CTBeHHOMY qHCJIY k2 = 1 + 2i OTBeqaeT KOMnJIeKCHbIft c06cTBeHHbdl BeKTOp, nOJIyqaeMbIft H3 CHCTeMbI
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HJIH |
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AHaJIOrHqHO .n;JIf! k3 = 1 - |
2i: HMeeM |
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1 |
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2iC1 + C2 = 0, |
{C1 = 1, |
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-2i |
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{ -C1 + 2iC2 - C3 = 0, => |
C2 = -2i, |
HJIH |
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3C2 - 2iC3 = ° |
C3 = 3 |
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06rn;ee pemeHHe CHCTeMhl MO}l{HO 3anHcaTb B BH.n;e |
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OCTaJIOCb nOKoop.n;HHaTHO B3f!Tb OT npaBoft qacTH .n;eftcTBHTeJIbHYK> qacTb:
x = |
~et + ~etcos 2t + ~C3etcos 2t, |
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y = |
~et sin 2t - ~et sin 2t, |
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z = - ~C1et + ~C2etcos2t+ ~C3etcos2t.
Pew:umb CUCmeM'IH ypaBHeHuiJ. (Bce rfiyH'~'ll,UU apeYMeHma t):
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X" = y, |
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dY |
= xy, |
2.8.20. |
2.8.21. |
{ dt |
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dz |
dy |
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y" =x. |
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dt |
+ dt = z + xy. |
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X' = X -y +z, |
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2.8.22. |
{ y' =x+y-z, |
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z' = -y + 2z. |
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Pewumb CUCmeM'bt ypa6HeHui1.:
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dy -1-! |
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dY |
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{ dx - |
z' |
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{ dx = yZ' |
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2.8.23. |
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2.8.24. |
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dx |
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dY . |
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y-z, |
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- =SlllX- |
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x-y |
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2.8.25. |
dx |
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2.8.26. |
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~~ = COS X + 4y + |
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z - t' |
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2z. |
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dz =x-y+l. |
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dz |
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2.8.27. |
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KOHTponbHble BOnpOCbl M 60nee cnO>KHble 3aAa'iM
2.8.28. ECTb JIH pa3HHD;a B 3anHCH co6cTBeHHblx BeKTopoB MaTpHD;bI B 06- m;eM BH.IJ;e HJIH B HOpMHpoBaHHoM BH.IJ;e?
Pewumb CUCmeM'bt ypa6HeHui1.:
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dx |
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x - y + z, |
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dt |
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2.8.29. |
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dy |
= -x + 5y - |
z, |
2.8.30. |
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dz |
= x - y + 3z. |
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2.8.31. |
{dy~~ = 3x+5y, |
npH YCJIOBHH x(O) |
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dt |
= -2x - |
8y. |
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d |
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4y + 1 + 4t |
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~ = - 2x - |
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2.8.32. |
{ |
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2.8.33. |
d |
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.J!.. = -x + y + ~t2 |
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dt |
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2 . |
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dx = 2x + y - 2z - t + 2 |
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2.8.34. |
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= 1- x |
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dt |
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~~ =x+y-z-t+l.
2.8.35. |
dt |
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dx _ |
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x 2 _ y2 |
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2tx - |
2ty· |
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t 2 _ |
= 2, y(O) = 5.
{XI = 3x + 4y + 2z,
y' = x + 4y + z,
Zl = 4x + 6y + 5z.
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2.8.36. { t'dY = (tx + ty + 2x - t) dt, t . dx = (t - 2x) dt.
KOHTPOl1bHAH PA60TA
BapMaHT 1
1. HaiiTH 06m;ee peweHHe ,ll;Hq,q,epeHII.HaJIbHoro ypaBHeHHH
(1 + X 2)y" + 2xy' = 7x3.
2. PewHTb 3a,n;aqy KOWH:
y" . y3 = 4y4 - 0,25, |
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y(O) = .;2' y'(O) = V;. |
3. HaiiTH o6m;ee peweHHe ,ll;Hq,q,epeHIJ;HaJIbHOrO ypaBHeHHH y" - 3y' + 2y = (4x + 9)e2x .
4. HaiiTH 06m;ee peweHHe ,ll;Hq,q,epeHII.HaJIbHOrO ypaBHeHHH y" + 6y' + 13y = e-3x sin 2x.
5. HaiiTH peweHHe 3a,n;aQH KOWH:
y" + 4y = Sin\x' y (i) = 2, y' (i) = 11'.
6. PewHTb CHCTeMY ,ll;Hq,q,epeHIJ;HaJIbHbIX ypaBHeHHii
X' = X + y + 3et ,
{
y' = 2x - y + cos 2t.
BapMaHT 2
1. HaiiTH o6m;ee peweHHe ,ll;Hq,q,epeHIJ;HaJIbHOrO ypaBHeHHH
X 4 y" + x 3 • y' = 10.
2. PewHTb 3a,n;aQy KOWH: |
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y"y3 + 4 = 0, |
y(l) = 2, |
y"(l) = 2. |
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HaiiTH o6m;ee peweHHe ,ll;Hq,q,epeHIJ;HaJIbHOrO ypaBHeHHH |
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y" + 3y' + 2y = (1 - |
2x)e- x . |
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4. HaiiTH 06m;ee peweHHe ,ll;Hq,q,epeHIJ;HaJIbHOrO ypaBHeHHH |
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y" + 4y' - 5y = 2x3e- 2x sin3x. |
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5. |
HaiiTH peweHHe 3a,n;aQH KOWH: |
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y" - 3y' + 2y = 3 +le_ x ' |
y(O) = 1 + 81n2, y' = 71n4. |
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6. |
PewHTb CHCTeMY ,ll;Hq,q,epeHIJ;HaJIbHbIX ypaBHeHHii |
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X' |
= 2x + 3y - |
et , |
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{ |
= X - 3y - |
sin 2t. |
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y' |
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BapMaHT 3
1. HaitTH 06m;ee pemeHHe .II.H<p<pepeHII.Ha.JIbHoro ypaBHeHHf!
(x 2 + 1) . y" + 2xy' = x(x2 + 1).
2. PemHTb 3a,n,a'lYKomH:
y" = 8sin3 ycosy, y(1) =~, y'(1) = l.
3. HaitTH o6m;ee pemeHHe ,ll;H<p<pepeHI.J;Ha.JIbHoro ypaBHeHHf! y" - 2y' - 3y = (8x + 4)e- x.
4. HaitTH o6m;ee pemeHHe ,ll;H<p<pepeHI.J;Ha.JIbHoro ypaBHeHHf! y" - 8y' - 9y = (2x + 1)e4x sin5x.
5. HaitTH pemeHHe 3a,n,a'lHKomH:
y" - 3y' = |
-3x |
,y(O) = In4, y'(O) = 31n4 - l. |
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3 |
+ e- 3x |
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6. PemHTb CHCTeMY ,ll;H<p<pepeHI.J;Ha.JIbHbIX ypaBHeHHit
X' = 3x - 2y + e-t ,
{
y' = 5x + 6y - 3 sin t.
BapMaHT 4
1. HaitTH o6m;ee pemeHHe ,ll;H<p<pepeHII.Ha.JIbHoro ypaBHeHHf! x4y" + x 3 y' = 5.
2. PemHTb 3a,n,a'lYKomH:
y" = 50sin3 ycosy, y(1) =~, y'(1) = 5.
3. HaitTH 06m;ee pemeHHe ,ll;H<p<pepeHI.J;Ha.JIbHOrO ypaBHeHHf! y" + 2y' - 3y = (x2 + 2x - 3)ex.
4. RaitTH o6m;ee pemeHHe ,ll;H<p<pepeHI.J;Ha.JIbHOrO ypaBHeHHf! y" + 4y' + 8y = (x + 2)e-2X cos 3x.
5. RaitTH pemeHHe 3a,n,a'lHKomH:
Y" + 3y' + 2y = |
1 |
1 |
y |
(0) 0 |
y'(O) = O. |
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+ 2ex ' |
=, |
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6. RaitTH o6m;ee pemeHHe JIHHeitHoit CHCTeMhl ,ll;H<p<pepeHI.J;Ha.JIbHbIX ypaBHe-
HHit |
y' + 2x - 3y - 3e2t |
= 0, |
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{ |
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x' + x + 4y |
+ cos t |
= O. |
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BaplilaHT 5
1. HaRTH o6m:ee peweHHe ,ll;H<p<pepeHll,HaJIbHOro ypaBHeHHjI
(1 + x2)y" + 2xy' = x 3 + x.
2. PewHTb 3a,zr.aQy KOWH:
y" = 16y3, y(4) = 1, y'(4) = 4.
3. HaRTH o6m:ee peweHHe ,ll;H<p<pepeHll,HaJIbHOrO ypaBHeHHjI
y" - 4y' + 4y = (2X2 - 3x + 2)e2z •
4. HaihH o6m:ee peweHHe ,ll;H<p<pepeHll,HaJIbHoro ypaBHeHHjI
y" - 2y' + 5y = (2x - 3)eZ cos 2x.
5. HaRTH peweHHe 3a,zr.aQH KOWH:
,,1 |
, |
y -y= 1+2ez ' |
y(0)=31n3, y(0)=21n3-1. |
6.HaRTH o6m:ee peweHHe CHCTeMbI ,ll;H<p<pepeHll,HaJIbHbIX ypaBHeHHii
{X' + y' + 2x - 3y - et = 0,
2x' - 3y' + x - 2y + cos t = O.
o
rnaBa 3. KPATHblE lIIHTErPAllbl
§1. ABOli1HOli1 VlHTErPAll. CBOli1CTBA VI METOAbl Bbl'-lVlCllEHVlH
Onp~eneHllle III reOMeTplII'"IeCKllliiiCMblcn ABoiiiHoro l'IHTerpana
I1YCTb D - HeKOTopaH 3aMKHYTaH 06JIacTb B rrJIOCKOCTH Oxy, Ha KOTOPOil: onpe,D;eJIeHa HerrpepbIBHaH <PyHKIJ;HH ,D;ByX rrepeMeHHbIX Z = f(x, y). Pa306beM 06JIaCTb D Ha n «3JIeMeHTapHbIX 06JIaCTeil:» Di (i = r,n), IIJIoru;a,D;H KOTOPbIX 0603Ha'lHMCOOTBeTCTBeHHO 'Iepe3!:l.Si. Terrepb B K8JK,D;Oil: 06JIaCTH Di BbI6epeM rrpon3BOJIbHYIO TO'lKYMi(Xi, Yi) (pHC. 9), rrOCJIe 'IemCOCTaBHM CyMMy
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II f(x,y)dS HJIH |
II f(x,y)dxdy. |
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B aTOM CJ1Y'IaerOBopHT, 'ITO<PYHKIJ;HH !(X, y) U'HmeepupyeMa Ha 06JIaCTH D. TIPll
aTOM <PYHKIJ;HH !(X, y) Ha3bIBaeTCH noiht'Hmeepa.!/!b'HoiJ, IjjY'H'lC'Il,ueiJ" a |
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ECJ1H <PYHKIJ;HH !(x, y) HerrpepbIBHa B 06JIacTH D, TO OHa HHTerpHPyeMa. |
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TeopeMa 3.1. Ecnlll !(x, y) ~ 0 III HenpepblBHa B 06naCTIII D, TO IIIHTerpan |
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Bblpa>KaeT 06beM Tena, OrpaHIII'leHHOrOCHIII3Y 06naCTbIO D, cBepxy - |
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B aTOM 3aKJIIO'IaeTCHreOMeTpH'IeCKHiiCMbICJI ,n:BoiiHoro HHTerpaJIa.
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y D
Puc. 10
B '1acTHOCTH,eCJ1H !(X, y) == 1, TO II !(x, y) dS paseH IlJIOID;a,n:H 06JIaCTH D:
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S(D) = II dS = II dxdy.
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CBoiicTBa ,n:BoiiHoro HHTerpaJIa aHaJIOrH'IHbI cooTBeTcTBYIOID;HM cBoiicTBaM orrpe,n:eJIeHHOrO HHTerpaJIa.
128
1. JIu:He11:Hocmb. ECJIH <PYHKIJ;HH I(x, y) H g(x, y) HerrpepblBHbI Ha 06JIacTH D,
TO II(a. I(x,y) ±(3. g(x,y)) dxdy = a· II I(x,y)dxdy+ (3. II g(x,y)dxdy
D D D
(0: H (3 - rrOCTOHHHble 'IRCJIa). B'IacTHOCTH,
IIa/(x, y) dxdy = a II I(x, y) dxdy,
D
T. e. rrOCToHHHbliI: MHO)KHTeJIb MO)KHO BblHOCHTb 3a 3HaK ,n:Boil:Horo HHTerpaJIa.
2. MO'HOmo't/:Hocmb. ECJIH <PYHKIJ;HH I(x, y) H g(x, y) HerrpepblBHbI Ha 06JIacTH D II BCIO,LJ:y B 3TOil: 06JIacTH I(x, y) ~ g(x, y), TO
II I(x,y)dxdy ~ II g(x,y)dxdy.
D D
TaKHM 06pa30M, HepaBeHCTBa MO)KHO rrO'lJIeHHOIIHTerpHpOBaTb. B '1aCTHOCTII,eCJIII m ~ I(x,y) ~ M, V(x,y) ED, TO
m· S ~ II I(x,y)dxdy ~ M· S,
D
r,l.l;e S = S(D) - rrJIOIIl;a,LJ:b 06JIaCTH D. )J;aHHble HepaBeHCTBa Ha3b1BaIOTCH otje'H"lCoit u'ltmet!pa.!la. EIIl;e O,n:HO CJIe,n:CTBHe: eCJIH I(x, y) ~ 0 Ha 06JIaCTH D, TO
II I(x,y)dxdy ~ O.
D
3. TeopeMa 0 CpeO'HeM 3'1ta"te'HUu.
TeopeMa 3.2. ECI1~ c\>YHKLI~ll I(x, y) HenpepblBHa Ha 0611aCT~ D, TO CYUlecTByeT TOYKa Mo(xo, YO) E D TaKall, YTO
II I(x, y) dxdy = I(xo, YO)· s, |
~11~ ~II I(x,y)dxdy = I(xo,yo). |
D |
D |
IIpH 3TOM 3Ha'leHHeI(xo, yo), T. e. '1HCJIO
~111(x,y)dXdy,
D
lfa3bIBaeTCH U'Hmet!paJlb'lt'btM CpeO'HUM 3Ha'leHHeM<PYHKIJ;HH I(x, y) B 06JIaCTH D.
4. AOOUmU6'1tOCmb. ECJIH 06JIaCTb D rrpe,n:cTaBJIHeTCH B BH,n:e 06'be,n:HHeHHH,n:BYX
06JIaCTeil: Dl H D2 6e3 06IIl;HX BHYTpeHHHx TO'leK,TO
II I(x,y)dxdy = II I(x,y)dxdy+ 111(x,y)dXdy.
D |
Dl |
D2 |
5 C60pHHK 38,lUl'l no BblcweA M8TeM8THKe. 2 K)'pC |
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129 |