1. Интегралы ФНП Диф_ур (853736), страница 31
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(19.1) y = u v y0 = u0 v + uv0 : A (19.1) u0 v + uv0 + p(x)uv = q(x):T, , u0v + uNv0 + p(x)v] = q(x):) u(x) v(x) ( 0v + p(x)v = 0(19:9)0u v = q(x):(19:10)5 v 6 0 (19.9), (19.10) u(x) .) - y = uv: 19.3. J y0 ; y th x = ch2 x:< V:) y = u v y0 = u0 v + uv0 u0v + uNv0 ; th xv] = ch2 x:) u(x) v(x) ( 0v ; th xv = 0(19:11)20u v = ch x:(19:12)' v0 ; th xv = 0 :dv = th xv dv = th xdx Z dv = Z sh x dxdxvvch x484Z d(ch x)ln jvj = ch x ln jvj = ln j ch xjv = ch x { (19.11). = (19.12) :u0 ch x = ch2 x u0 = ch x:Z7 : u = ch xdx u = sh x + C:) : y = (sh x + C ) ch x: > 19.4.
J 1dy =dx x cos y + sin 2y :< dx = x cos y + sin 2y dx ; x cos y = sin 2y:dydy , x y (x = x(y)). 4 V ( x y).0 = u0 v + uv 0 ) x = u v u = u(y) v = v(y): = dx=xdy u0 v + uNv0 ; v cos y] = sin 2y:) u(y) v(y) ( 0v ; v cos y = 0u0v = sin 2y:' v0 ; v cos 2y = 0 :dv = v cos y dv = cos ydy Z dv = Z cos ydydyvvln jvj = sin y v = esin y { . u0 v = sin 2y u0esin y = sin 2y u0 = sin 2ye; sin y :4857 :=2u=Z= ;2ZZsin 2ye; sin y dy = 2sin ye; sin y d(sin y) = Ntd(e;t) =hZsin ye; sin y cos ydy =Z t = sin y] = 2 te;tdt =Z ;t ; e dt = ;2(te;t + e;t) + C =;2= ;2e; sin y (sin y + 1) + C:te;ti) x = uv = ;2e; sin y (sin y + 1) + C esin y x = Cesin y ; 2(sin y + 1): > &,$ .1) xy 0 2y = 2x4;y + y tg x = sec x4) y = x(y ; x cos x)y6) y ; = xx2);3) (xy + ex)dx xdy = 05) 2x(x2 + y )dx = dy 000y + 2xy = x3y ; 1 ; x = 0 y = 0 * x = 08) y ;1 ; x21 y = 0 * x = 09) y ; y tg x =cos x10) xy + y ; ex = 0 y = b * x = a:7)0000y = Cx2 + x43) y = ex (ln jxj + C ) x = 025) y = Cex ; x2 ; 1x4 + C 7) y =6x2x 9) y =cos xy = sin x + C cos x4) y = x(C + sin x)6) y = Cx + x21)2)p1= (x 12; x2 + arcsin x)x a b ; eae10) y =+:s8)y|||||486xx1+x1 x;% .A y0 + p(x)y = q(x)y ( 6= 0 6= 1)(19:13) , .J V , V (y = u v): 2 , 9 y = 0: 19.5.
J xy0 + y = y2 ln x:< x 6= 0 (19.13):y0 + x1 y = lnxx y2(19:14) p(x) = 1=x q(x) = ln x=x = 2: . 2y y0 + x1 y = 0 :dy = ; 1 y dy = ; dx Z dy = ; Z dx dxx yxyx ln jyj = ; ln jxj + ln jC j ln jyj = ln Cx y = C=x { . - (19.14) 00y = C x(x) y0 = C (x)xx;2 C (x) = C x(x) ; Cx(2x) : y y0 (19.14), C 0 (x) ; C (x) + C (x) = ln x C 2(x) @xx2x2x x2487C 0(x) = C 2(x) lnx2x { . dC = C 2 ln x dC = ln x dx Z dC = Z ln x dx:dxx2 C 2 x2C2x2: : { , { Z dCZ ln xZ1C 2 = ; C (x) x2 dx = ln xd(;1=x) =ZZ= ; lnxx + x1 d(ln x) = ; lnxx + x12 dx = ; lnxx ; x1 + C:= , (C ! ;C ); C (1x) = ; lnxx ; x1 ; C C (x) = 1 + Cxx + ln x y = C (x)=x y = 1 + Cx1 + ln x : = y = 0 { y, C :()1y = 1 + Cx ln x y = 0 : > 19.6.
J ' Vy0 + 31 y = 13 y12 y(0) = 1:< - V(y = uv y0 = u0v + uv0 ) u0 v + uv0 + 13 uv = 31 u21v2 "# 1 1100u v + u v + 3 v = 3 u2 v 2 :4884 u(x) v(x) 8>> v0 + 1 v = 0<3>0>: u v = 12 2 :3u v v0 = ; 31 v v = e;x=3: , xe100;x=3u e = 3u2e;(2x)=3 u = 3u2{ 9 .
ZZp2x23u du = e dx 3u du = exdx u3 = ex + C u = ex + C:D y = uv qppy = ex + C e;x=3 = (ex + C )e;x = Ce;x + 1:p= , y = Ce;x + 1 { . 4 ' x = 0 y = 1:p1 = Ce0 + 1 C = 0:-: y = 1: >333333 &,$ .y xy21) y 0 =3)y0;x ;+ 2y = y 2 ex ;2) 2xy y 0 y 2 + x = 04) xy 2y 0 = x2 + y 3:1) y (x2 + Cx) = 13) y (ex + Ce2x) = 1C2) y 2 = x ln x4) y 3 = Cx3 3x2:|||||489; 20 A P (x y)dx + Q(x y)dy = 0(20:1) , u = u(x y) ..@u dx + @u dy:Pdx + Qdy du @x@y: 9 (20.1) du = 0 9 u(x y) = C:+ ) , (20.1) , @P @Q :@y @x) 5 u(x y) , du = 0 ..@u = P @u = Q:(20:2):@x@y7 (20.2) x, y , Zu = P (x y)dx + '(y)(20:3) '(y) { y.4 (20.3) y:@u = @ Z P (x y)dx + '0(y)@y @y @u=@y (20.2):@ Z P (x y)dx + '0 (y) = Q(x y)@y '0 (y) y '(y). '(y) (20.3), u(x y):490) - u(x y) = C: 20.1.
J (sin xy + xy cos xy)dx + x2 cos xydy = 0:< ) F P (x y) = sin xy + xy cos xy Q(x y) = x2 cos xy@P = (sin xy + xy cos xy)0 = x cos xy + x cos xy ; x2y sin xy =y@y= 2x cos xy ; x2y sin xy@Q = (x2 cos xy)0 = 2x cos xy ; x2y sin xy:x@x@Q , , = , @P@y @x.) 4 u(x y) 8>@u = sin xy + xy cos xy>>< @x>@u = x2 cos xy:>>: @y7 x:ZZZu = (sin xy+xy cos xy)dx+'(y) = sin xydx+ xy cos xydx+'(y):: , y { :ZZ1sin xydx = y sin xyd(xy) = ; y1 cos xyZZZxy cos xydx = y x cos xydx = y x y1 d(sin xy) =ZZ= xd(sin xy) = x sin xy ; sin xydx = x sin xy + y1 cos xy( ). = ,u = ; y1 cos xy + x sin xy + y1 cos xy + '(y) 491u = x sin xy + '(y):(20:4)4 y:@u = (x sin xy + '(y))0 = x2 cos xy + '0 (y)y@y @u=@y :x2 cos xy + '0 (y) = x2 cos xy '0(y) = 0Z '(y) = 0dy = 0 { .
'(y) = 0 (20.4) u(x y) :u(x y) = x sin xy:) - u(x y) = C ) x sin xy = C: >F, , (20.1) du = 0 ) .J 9 . 20.2. J (2x + y)dx + (x + 2y)dy = 0:< ) F P (x y) = 2x + y Q(x y) = x + 2y@P = (2x + y)0 = 1 @Q = (x + 2y)0 = 1:yx@y@x@Q , , = , @P@y @x.) 4 u(x y) 2xdx + 2ydy + (ydx + xdy) = 0:= 2xdx = dx2 2ydy = dy2 ydx + xdy = d(xy) dx2 + dy2 + d(xy) = 0492 d(x2 + y2 + xy) = 0:5 u(x y) = x2 + y2 + xy , du = 0: -:x2 + y2 + xy = C: > 20.3.
J (3x2y + y3)dx + (x3 + 3xy2 )dy = 0:< ) F P (x y) = 3x2y + y3 Q(x y) = x3 + 3xy2 @P = (3x2y + y3 )0 = 3y2 + 3x2 @Q = (x3 + 3xy2 )0 = 3x2 + 3y2:yx@y@x@Q , , = , @P@y @x.) 4 u(x y) (3x2ydx + x3dy) + (3xy2 dy + y3dx) = 0:= 3x2ydx + x3dy = d(x3y) 3xy2 dy + y3dx = d(y3x) d(x3 y) + d(y3x) = 0 d(x3y + y3x) = 0:5 u(x y) = x3y + y3 x , du = 0: -:x3y + y3x = C: >$*' @Q =J , (20.1) @P@y @x, = (x y), , , Pdx + Qdy = 0 .J : e (20.1) @P=@y ; @Q=@x = f (x)Q493 x = (x) R(x) = e f(x)dx:. (20.1) @P=@y ; @Q=@x = g(y)P y = (y) R g(y)dy;(y) = e: 20.4.
J (x cos y ; y sin y)dy + (x sin y + y cos y)dx = 0:< F P (x y) = x sin y + y cos y Q(x y) = x cos y ; y sin y@P = x cos y + cos y ; y sin y @Q = cos y:@y@x@P=@y ; @Q=@x = x cos y ; y sin y = 1:Qx cos y ; y sin y9 R 1dx = (x) = e = ex:A ex ex(x sin y + y cos y)dx + ex(x cos y ; y sin y)dy = 0{ .4, ( ) u(x y) 8>@u>< @x = ex(x sin y + y cos y)>@u>: = ex(x cos y ; y sin y):@x4947 x, Z xZ xu = e (x sin y + y cos y)dx + '(y) = sin y e xdx+Z+y cos y exdx + '(y) = exx sin y ; ex sin y + exy cos y + '(y): ()4 u(x y) y @u = (exx sin y ; ex sin y + exy cos y + '(y))0 =y@y= exx cos y ; ex cos y + ex cos y ; exy sin y + '0 (y): @u=@y '0(y) = 0 ) '(y) = 0 { : '(y) = 0 (), u(x y) :u = exx sin y ; ex sin y + exy cos y:-: ex(x sin y ; sin y + y cos y) = C: > 20.5.
J ydx ; (x + y2)dy = 0:< F P (x y) = y Q(x y) = ;(x + y2)@P = 1 @Q = ;1@y@x@P=@y ; @Q=@x = 2 :Py9 R 2=y dy; = (y) = e= e; lny = eln(1=y ) = 1=y2:A 1=y2 y dx ; x + y2 dy = 0y2y249522{ .? u(x y) , du = 0 . :ydx ; xdy ; dy = 0:y2= ydx ; xdy = d x ! dy = d(y)y2y!xd y ; y = 0 x ; y = C x ; y2 = Cy:()yA, 1=y2 y = 0 (**): fx ; y2 = Cyy = 0g: > &,$ .1) (2x + 3x2 y )dx + (x3 3y 2)dy = 03) (2 9xy 2)xdx + (4y 2 6x3)ydy = 0y5) dx + (y 3 + ln x)dy = 0;;;2) 2xydx + (x2e y dx ; (2y + xe y )dy = 06) (3x2 + 6xy 2 )dx + (6x2 y + 4y 3 )dy = 08) (x2 + y 2 + 2x)dx + 2xydy = 0y10) dx ; (4xy + 1)dy = 0:x4)x7) (x + y )dx + (x + 2y )dy = 03!y29) 2xy + x y +dx+ (x2 + y 2)dy = 03;1) x2 + x3y y 3 = C 3) x2 3x3y 2 + y 4 = C ;5) 4y ln x + y 4 = C x2 + xy + y2 = C 7)22!yx29) ye x += C3; y2)dy = 0;; y3 = C 4) xe y ; y 2 = C 2) 3x2 y;6) x3 + 3x2 y 2 + y 4 = C x3 + xy2 + x2 = C 8)3y10) + 2y 2 = C:x|||||496; 21 , % y (n) = f (x)- y(n) = f (x) n{ .
5.1. J y000 = sin x + x:< 3 , 2Zx00y = (sin x + x)dx = ; cos x + 2 + C1@32Zxx0y = (; cos x + 2 + C1)dx = ; sin x + 6 + C1x + C2@423Zxxxy = (; sin x + 6 + C1x + C2)dx = cos x + 24 + C1 2 + C2x + C3 C1 C2 C3 { . = C21 { , :4xy = cos x + 24 + C1x2 + C2x + C3: > 21.2. J ' 8>< y000 = ln x (21:1)2x>: y(1) = 0 y0 (1) = 1 y00 (1) = 2:(21:2)< ( ) (21.1):Z ln xZZ 1 11100y = x2 dx = ln xd(; x ) = ; x ln x + x x dx =497Z= ; x1 ln x + x12 dx = ; x1 ln x ; x1 + C1:5 C1, (21.2): y00 = 2 x = 1: :2 = ;11 ln 1 ; 11 + C1 2 = 0 ; 1 + C1 C1 = 3 , ,y00 = ; x1 ln x ; x1 + 3: ZZZZy0 = (; x1 ln x ; x1 + 3)dx = ; ln xd(ln x) ; x1 dx + 3dx == ;21 ln2 x ; ln x + 3x + C2:5 C2, (21.2): y0 = 1 x = 1.1 = ; 12 ln2 1 ; ln 1 + 3 + C2 1 = 0 + 3 + C2 C2 = ;2 , ,y0 = ;21 ln2 x ; ln x + 3x ; 2: ZZ ln2 xdx Zy = ; 2 ; ln xdx + (3x ; 2)dx:: ,y = ; x2 ln2 x + 32 x2 ; 2x + C3:(21:3)5 C3, (21.2): y = 0 x = 1:0 = ; 12 ln2 1 + 23 12 ; 2 + C3 0 = 0 + 32 ; 2 + C3498 C3 = 1=2: o C3 = 1=2 (21.3), 2 x 3x2lny = ; x + 2 ; 2x + 12 :(= x = 1 > 0 x1 ln x.) >%, ' * * *- F (x y(k) y(k+1) ::: y(n)) = 0(21:4) y(x) (k ; 1){ .+ ) y(k) = z (x), z (x) { , (21.4) k , ..
F (x z z 0 z 00 ::: z (n;k)) = 0:(21:40)) O 9 , .. z x (n ; k) :z = f (x 1 2 ::: n;k) z = y(k) (21.1):y(k) = f (x C1 C2 ::: Cn;k):) 5 y k . 21.3. J xy(4) ; y000 = 0:< ) A y y0 y00 , 9 y000 = z (x), y(4) = (y000 )0 = z 0(x). = , o xz 0 ; z = 0:499dz ; z = 0 { .) x dx4 U 18, :dz = z xdz = zdx dz = dx Z dz = Z dx x dxz xzxln jz j = ln jxj + ln jC1j ln jz j = ln jC1xj z = C1x:= z = y000 y000 = C1x:) = :2Zy00 = C1xdx = C1 x2 + C213Z 0 x2C1x0@Ay = C1 2 + C2 dx = 6 + C2x + C3@1Z 0 C1x3y = @ 6 + C2x + C3A dx = C241 x4 + C22 x2 + C3x + C4:= C1=24 C2=2 { , , C1 C2, = C1x4 + C2x2 + C3x + C4: > 21.4.
J ' 8 4 000< x y + 2x3y00 = 1: y(1) = 21 y0 (1) = 12 y00 (1) =;1:(21:5)(21:6)< ) A (21.5) y y0 , 9 y00 = z (x), y000 = (y00 )0 = z 0: = , (21.5) x4z 0 + 2x3z = 1:) = (21.6) x = 1 6= 0, , x4, (21:7)z 0 + x2 z = x14 500 9 . , , V (. U19).J z = uv, z 0 = u0v + uv0 . "# 1200u v + u v + x v = x4 :4 u v 82>>< v 0 + x v = 0>>: u0 v = 14 :x: :dv = ; 2 v dv = ; 2dx Z dv = Z ; 2dx dxxvxvx ln jvj = ;2 ln jxj ln jvj = ln x12 v = x12 : , :Z 111100u x2 = x4 u = x2 u = x2 dx u = ; x1 + C1:= z = uv, (21.7) !11z = ; x + C1 x2 z = C1 x12 ; x13 :A, z = y00 , (21:8)y00 = Cx21 ; x13 :) @ o (21.6).C1 : y00 = ;1 x = 1, (21:8):;1 = C121 ; 113 501 C1 = 0 , ,Zy00 = ; x13 y0 = ; x13 dx + C2 y0 = 21x2 + C2:C2 : 21 = 2 112 + C2, C2 = 0 , ,Z 110y = 2x2 y = 2x2 dx + C3 y = ; 21x + C3:C3 p : 21 = ; 2 1 1 + C3, C3 = 1.
= , y = 1 ; 21x : >%, ' * *-p F (y y0 y00 ::: y(n)) = 0 x.+ ) y0 = p(y), p(y) { . d (p(y)) = dp dy = dp py00 = dxdy dx dy! d dp ! dy!22pdpddpd2000y = dx dy p = dy dy p dx = p dy2 + p dy .. 9 y0 y00 ::: y(n) (n ; 1)- p(y).) - 9 , .. p(y):) F p(y) y0 , y(x). 21.5.
J 1 + (y0 )2 = yy00 :502< ) A , 9 dp p: y0 y00 y0 = p(y). O y00 = dy, dp p1 + p2 = y dy{ y .) :dp = 1 + p2 ypdp = (1 + p2)dy p dp = dy yp dy1 + p2yZ dy 1 Z d(p2 + 1)Z pdp=1 + p2y 2 p2 + 1 = ln jyj + ln jC1jln jp2 + 1j = 2 ln jC1yj ln jp2 + 1j = ln jC12y2j p2 + 1 = C12y2q 2222 p = C1 y ; 1 p = C1 y2 ; 1:) F p y0 , dy = qC 2y2 ; 11dx{ p ( "+" ";").
J :ZZdydyq 2= dx q 2 2 = dxC1 y2 ; 1C1 y ; 11 ln(C y + qC 2y2 ; 1) = (x + C ) 121C1qln(C1y + C12y2 ; 1) = (x + C2)1: > 21.6. J ' y00 y3 + 1 = 0y(1) = ;1 y0 (1) = ;1:< ) A , 9 y0 = p(y)dp p: = y00 = dydp py3 + 1 = 0:dy503) :dp py3 = ;1 py3dp = ;dy pdp = ; dy dyy3ZZp2 = 1 + C1 p2 = 1 + C pdp = ; dyy 3 2 2y 2 2y2 1vut 1 + C1 :p = uy2) F p y0 , :vudy = ut 1 + C1:dxy2 y = ;1 y0 = ;1, 1 :;1 = p1 + 1:: ";", 0 , ,;1 = ;p1 + C1 C1 = 0 vudy = ;ut 1 dy = ; 1 :dxy2dxjyj: y(1) = ;1 < 0, 9jyj = ;y :dy = 1 dx y2ZZyydy = dx ydy = dx 2 = x + 2:7 :(;1)2 = 1 + C = ; 1 :2222504C,y2 = x ; 1 y2 = 2x ; 1 y = p2x ; 1:22: y(;1) = ;1 < 0 p ";".
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