1. Интегралы ФНП Диф_ур (853736), страница 30
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16.3).;. 16.35 :@z = 2x + 4y ; 12 @z = 4x ; 4y:@x@y7 ( 9)8< x + 2y ; 6 = 0: x;y = 0 x = y = 2:= M1(2 2) D.- , 9.5 AC , .. x = 1, z = ;2y2 + 4y ; 1:5 zy0 = ;4y + 4 = ;4(y ; 1)463zy0 = 0 y = 1:2, M2(1 1), A, 9.5 AB , .. y = 1 z = x2 ; 8x +8:5 zx0 = 2x ; 8 = 2(x ; 4)zx0 = 0 x = 4:2, M3(4 1) 9.5 CB , .. y = 7 ;2 x 23xz = ; 2 + 9x ; 292:5 zx0 = ;3x + 9zx0 = 0 x = 3:2, M4(3 2) 9.: M1 M2 M3 M4 B C:z (2 2) = ;2 z (1 1) = 1 z (4 1) = ;8z (3 2) = ;1 z (5 1) = ;7 z (1 3) = ;7:7, z = 1 z = ;8: > ;1) ) !#$, $, 67 z = x(y 1) + x2+y 2 1:2) ) !#$, $, 67 z = x2 + y 2 4x + 4y + 1 +#$ , A(1 0) B (4 0) C (1 3):p3 3*1) z =4;px=;2) z = 16 4 14 *p32y=;x=2+p142;1 z =2y =2;||||{464;pp3 33* x = y=4214 z = 7 * x = 22;p;; 21 :y = ;2: 17 - * F (x y y0 ) = 0 (a b) y = y(x), 9 y0 (x), x 2 (a b). A G(x y) = 0, , , , G(x y) = 0, .
. 17.1. 4 , y = sinx x xy0 + y = cos x.< y0 = (x cos xx;2 sin x) . 0 :2 cos x x sin x sin xsinxxxcosx;sinxx+ x = x2 ; x2 + x =x2= cos x ; sinx x + sinx x = cos x: cos x cos x, . > y = y(x) # 667#$ '.21) y = xe x xy = (1 ; x)y 2) y = xe x =2 xy = (1 ; x2 )y pp4) y = x 1 ; x2 yy = x ; 2x3 3) y = ; x4 ; x2 xyy = y 2 + x42 $, 67;0;000||||{ 17.2.
, y = e2x y0 = 2y C: C1 = 0 C2 = ;1 C3 = 1.< 5 y0 = (e2x)0 = (e2x)0 = 2e2x: 0 2Ce2x = 2Ce2x@e2x e2x:465; 1 = 0 y 0@ 2 = ;1 y = ;e2x@ 3 = 1 y = e2x:C . >. 17.1 17.3. y0 = 4x3. 5 , y(0) = 1, , M0(0@ 1).< A y0 = f (x) .74ZZx33y = 4x dx = 4 x dx = 4 4 + := , y = x4 + , C { .5 C = 1 = 0 : 1 = 04 + C C = 1: C = 1 , y = x4 + 1.T y = x4 + 1 { . >;. 17.2 4+$ 667#$ y 0 = f (x). ) ,, # '( #$ #' y (x0) = y0.466y3) y5) y1)000=; sin x y() = 1y = xex y(0) = ;124) y = xex y (0) = 1=22)= ln x y (1) = 01= y(0) = 0:1 + x200y = cos x + 224) y = ex =2 1)y = (x ; 1)ex5) y = arctg x:2)3)y = x ln x ; x + 1||||{, y 0 = f (x y ) .
5 y0 = f (x y), y(x0 ) = y0, .. , M0(x0 y0).4 ' D OXY f (x y) fy0 (x y).. : ' y0 = f (x): 17.4. , y(x0 ) = y0, ' 1) y0 = ye;x @2) y0 = sin(2y + x):2< 4 , OXY :1) f (x y) = ye;x (x@ y). fy0 = (ye;x )0y = e;x , je;x j = e;x 1.2) f (x y) = sin(2y + x) (x@ y).fy0 = 2 cos(2y + x) (x@ y), j2 cos(2y + x)j 2: >22224672 4 $, * #'!0 #$0 # 0 y (x0) = y0 ( , , # 667#$0 .1) y 0 = sin y cos x2) y 0 = arctg(x y )3) y 0 = arcctg(2x + 4y )4) y 0 = cos x2 + sin 3y:;||||{, 17.5.
5 , (0@ ;2), 9 9 , 3 .< y = y(x) . 9 y0 (x)@ (). 0 = 3 { 9 , (0@ ;2), ' y0 = 3y y(0) = ;2:J 0 = 3 { 9 , 9.= 9. = ,y0 = 3y , y = Ce3x. , ;2 = Ce0 C = ;2. - : y = ;2e3x: > 17.6. 5 , (1@ 1), N1 x] , 9, 2 ( ) (x > 0 y > 0).< Zx N1 x] y(t)dt.1 Zx1y(t)dt + 2 = 2xy:4684 9 (. ), y = 2(y + xy0 ), y0 = ; 2yx .
7 9 (. U18) (1) = 1, y = p1x : > 1) 2 $, , # *0 * ' M (x0 y0), $ <$.2) ) , # 0 *#($ +#$, ! + #$, $' !7, $ # * , 2.3) ) , # 0 +#$, *+ *( , $ # * , b. x)y = 2a23) b ln y ; y = x + C2) (C0<y< b:||||{469 18 A E (x)F (y)dx = G(x)H (y)dy(18:1) 9 , .+ *' ) J (18.1) e F (y)G(x), .. (18.1) E (x) dx = H (y) dy:G(x)F (y)) 7 Z H (y )Z E (x)G(x) dx = F (y) dy:) - (x y) = C(18:2)( y = (x C ) x = (y C )) C:. 4 F (y)G(x) e ,o 0 F (y)G(x).
= , (18.2) , .% *' 18.1. J 6xdx ; 6ydy = 2x2ydy ; 3xy2 dx:470< ) 2y(x2 + 3)dy = 3x(y2 + 2)dx (x2 + 3)(y2 + 2):2ydy = 3xdx :y2 + 2 x2 + 3 : dy , y dx { x:) : Z 2ydy Z 3xdxy2 + 2 = x2 + 3 . , , ( ):Z 2ydy Z d(y2 + 2)2 + 2)==ln(y22y +2y +2Z 2xdy3 Z d(x2 + 3) = 3 ln(x2 + 3) + C :=1x2 + 3 2 (x2 + 3) 22,ln(y2 + 2) = 32 ln(x2 + 3) + C12 ln(y2 + 2) = 3 ln(x2 + 3) + 2C1: , ln(y2 + 2)2 = ln(x2 + 3)3 + 2C12222ln (xy2 ++ 3)2)3 = 2C1 (xy2 ++ 3)2)3 = 2eC :- 2eC = C , (y2 + 2)2 = C: >(x2 + 3)347111 18.2. J y0 = 3y22x+ 1 :dy , < = y0 = dxdy = 2xdx 3y2 + 1 " "(3y2 + 1)dy = 2xdx . 7 ZZ(3y2 + 1)dy = 2xdxy3 + y = x2 + Cy3 + y ; x2 = C: > 18.3.
J x)(2;ex3e tg ydx + cos2 y dy = 0:< (ex ; 2) dy = 3ex tg ydxcos2 y tg y(ex ; 2):dy = exdxtg y cos2 y ex ; 2 ( ):Zdy = Z d(tg y) = ln j tg yjtg y cos2 ytg y472exdx = Z d(ex ; 2) = ln jex ; 2j + C:ex ; 2ex ; 2= ,ln j tg yj = ln jex ; 2j + C1Zln j extg;y 2 j = C1: tg y = eC :ex ; 2- eC = C , tg y = C tg y ; C (ex ; 2) = 0:(18:3)xe ;2 tg y(ex ;2) : y = n(n = 0 1 2 :::) { tg y = 0 x = ln 2 { ex ;2.
- y = n (18.3) C = 0, x = ln 2 (18.3) C: 2, . : ftg y ; (ex ; 2) = 0 x = ln 2g: > 18.4. 5 ' !0y sin x = y ln y y 2 = e:< ) 5 , :dy sin x = y ln ydxdy sin x = y ln ydx@ sin x y ln y:dy = dx :y ln y sin x11473 , Z dxZ dy=y ln ysin x :7 Z dxxxtg + ln jC j = ln C tg :=lnsin x22F ln jC j. = , y = ln x x > 0 ;1 +1. 7 :Z dyZ d(ln y)=y ln yln y = ln j ln yj:= , ln j ln yj = ln jC tg(x=2)j.
( C ):ln y = C tg(x=2) y = eC tg(x=2):5 y = eC tg(x=2):(18:4)) 5 C (18.4) x = 2 y = e, e = eC tg(=4) e = eC C = 1: (18.4) C = 1 y = etg(x=2): >. 4 dy = f (ax + by + d)dx a b d { , z = ax + by + d z = z (x) { , .474 18.5. J y0 = cos(x ; y):dy = 1 ; dz < 2 z = x ; y y = x ; z @ dxdx , dz = cos z1 ; dxdz = 1 ; cos z dz = (1 ; cos z )dx:dxJ , (1 ; cos z ) :dz = dx1 ; cos z , 1 ; cos z = 2 sin2 z2 ,Zdz = Z dx Z d(z=2) = Z dx:2 sin2(z=2)sin2(z=2) , C ; ctg z2 = x:F z x ; y x + ctg x ;2 y = C { .
> &,$ 667#$ :1)3)5)7)9)10)y = ; xy (ex + 1)dy ; yexdx = 0(2 + ex)yy = ex y = (x + y)2(2x + 3y ; 1)dx + (4x + 6y ; 5)dy = 0(2x ; y )dx + (4x ; 2y + 3)dy = 0:0004752)4)6)8)ppx 1 + y2 + yy 1 + x2 = 0y(1 + ln y) + xyq= 03(x2 y + y )dy + 2 + y 2 dx = 0y = (8x + 2y + 1)2 0001)3)5)7)9)10)p 2 q 22)y = Cx 1 + x + 1 + y = Cy = C (ex + 1)4) x(1 + ln y ) = C qy2 ; 2 ln(2 + ex) = C 6) 3 2 + y 2 + arctg x = C arctg(x + y ) = x + C 8) 8x + 2y + 1 = 2 tg(4x + C )x + 2y + 3 ln j2x + 3y ; 7j = C 5x + 10y + C = 3 ln j10x ; 5y + 6j:||||{- ? f (x y) n f (tx ty) tnf (x y).5, f (x y) = x2 + y2 + xy , f (tx ty) = (tx)2 + (ty)2 + (tx)(ty) = t2(x2 + y2 + xy) = t2f (x y): n = 0 . 5,22f (x y) = x2x2;+3yy2 { , 2(x2 ; 3y 2 )2 ; 3y 22 ; 3(ty )2(tx)txf (tx ty) = 2(tx)2 + (ty)2 = t2(2x2 + y2 ) = 2x2 + y2 = f (x y):4 dy = f (x y)(18:5)dx f (x y) .476+ ) dy = '( y ):dxx) 4 xy = z z = z (x) { dy = (xz )0 = z + xz 0 = z + x dz (18.5), y = xz dxdx !xzdz = '(z ) ; zdzz + x dz = ' x x dx 9 .) J , Zdz = Z dx'(z ) ; zx .) : z = y=x: 18.6.
J 2 + 2xy ; 5y 2x0y = 2x2 ; 6xy :< ) J x2(x2 6= 0 ), 21+2y=x;5(y=x)0y=2 ; 6(y=x) :dz ) 4 y=x = z y = xz y0 = z + x dx 2dz1+2z;5zz + x dx = 2 ; 6z 21+zdzx dx = 2 ; 6z :477) J (1 + z 2 6= 0 x 6= 0)(2 ; 6z )dz = dx1 + z2x Z (2 ; 6z )dz Z dx1 + z2 = x :Z dx= ln jxj + C { , x:Z dzZ 2zdzZ d(1 + z 2)Z 2 ; 6z1 + z 2 dz = 2 1 + z 2 ; 3 1 + z 2 = 2 arctg z ; 3 1 + z 2 == 2 arctg z ; 3 ln(1 + z 2) = 2 arctg z ; ln(1 + z 2)3@2 arctg z ; ln(1 + z 2)3 = ln jxj + C2 arctg z ; ln((1 + z 2)3jxj) = C:) F z y=x:2 arctg(y=x) ; ln((1 + y2=x2)3jxj) = C:A, 02 1322 322 3y@1 + A jxj = (x + y ) jxj = (x + y ) x2jxj6jxj5 2 + y2x2 arctg(y=x) ; ln jxj5 = C: > 18.7.
J xy0 = y + cos xy :< : (x 6= 0 )y0 = xy + cos xy 478dz @ y=x = z , y0 = z + x dx:dz = z + cos z x dz = cos z @z + x dxdx :dz = dxcos z x Z dzZ dxcos z = x( ):!zln tg 2 + 4 = ln jxj + ln jC j!zln tg 2 + 4 = ln jCxj: :!z(18:6)tg 2 + 4 = Cx: cos z z = 2 + n(n = 0 1 2 ::): n = 2k + 1 (18.6) C = 0 n = 2k (18.6) C: 9 (18.6) z = 2 + 2k.F z y=x, !!)(ytg 2x + 4 = Cx y = x 2 + 2k : > &,$ :2y + x1) y 0 =x2)y4790= ex=y +yx3)y5)(y7)9)0=y ; 1x; 2x)dx + xy dy = 02 y 2 y2y =x + 6 x + 332xy = 32yy2++42yxx2 4)y6)xy08)010)=0; x +y y qx2 + y 2 + y y = x2x+;2yy 2y2 :y = x x+2 ;xy2;xy0=001)3)5)7)9)x + y = Cx2y = x ln Cx x(y ; x) = Cyx + y = Cx(3x + y)sy 2 + y 2 = C x2x2)4)6)8)10)y = ;x ln ln Cx y = Cx ; x2 qy + x2 + y2 = Cx2qy2 arctg ; ln x2 + y 2 = C xyx2 + y2 = C:arctg ; lnxx||||{480 19 .
A y0 + p(x)y = q(x)(19:1) p(x) q(x) { , ( y y0 ).( + ) J y0 + p(x)y = 0(19:2) (. U18). . y = Cy0(x) y0 (x) 6 0:) - (19.1) y = C (x)y0 (x)(19:3) C (x) { . 5 y0 = C 0 y0 +Cy00 (19.1) y = C (x)y0 (x):C 0 y0 + Cy00 + p(x)Cy0 = q(x) C 0 y0 + C (y00 + p(x)y0) = q(x):(19:4)= y0 { (19.2), , , y00 + p(x)y0 0 (19.4) C 0y0 = q(x) 9 ():Z q(x)C (x) = y (x) dx + C:0 C (x) (19.3), (19.1).481 19.1. J y0 ; y ctg x = sin x:(19:5)< 7 , p(x) = ; ctg x q(x) = sin x:) 2 y0 ; y ctg x = 0: :dy = y ctg x dy = ctg xdx Z dy = Z ctg xdxdxyyln jyj = ln j sin xj + ln jC j ln jyj = ln jC sin xjy = C sin x { .) J y = C (x) sin x:(19:6) (19.5) y = C (x) sin x y0 = C 0 (x) sin(x)+C (x) cos x :C 0(x) sin x + C (x) cos x ; C (x) sin x ctg x = sin xC 0(x) sin x = sin x C 0 (x) = 1:7:ZC (x) = dx = x + C, (19.6), y = (x + C ) sin x: > 19.2.
J ' y0 cos2 x + y = tg x y(0) = 0:< ) J y0 cos2 x + y = 0: J , dy cos2 x = ;y dy = ; dx Z dy = ; Z dx dxycos2 xycos2 x482ln jyj = ; tg x+ln jC j ln j Cy j = ; tg x Cy = e; tg x y = Ce; tg x:) - y = C (x)e; tg x:(19:7) y = C (x)e; tg x y0 = C 0(x)e; tg x + C (x)e;tgx (;1= cos2 x)C 0(x)e; tg x cos2 x ; C (x)e;tgx + C (x)e; tg x = tg xC 0(x)e; tg x cos2 x = tg xZtg x etg xdx:C (x) = cos2x7 , :ZC (x) = tg xetg xd(tg x) = N t = tg x] =Ztetdt =Ztd(et) = tetZ; etdt = et(t ; 1) + C = etg x(tg x ; 1) + C:5 C (x) (19.7):hiy = etg x(tg x ; 1) + C e; tg x y = (tg x ; 1) + Ce; tg x { .) 4 ' y = 0 x = 0 C0 = (tg 0 ; 1) + Ce; tg 0 0 = C ; 1 C = 1: C = 1 , y = (tg x ; 1) + e; tg x: >483( .+ , ) J (19.1) y = u v(19:8) u = u(x) v = v(x) { .
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