1. Интегралы ФНП Диф_ур (853736), страница 14
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Z ; R a(x)dxR a(x)dx y = y(x6 C ) = eC+ eb(x)dx :(21:23)8 (21.23) /#0 (21.22) $ ". (21.21) / B.R a(x)dxL#0 (21.23)R 3Z ':/#0 y=eCo:o:R /#0 y:: = e a(x)dx e; a(x)dx b(x)dx: @( #( "3, /#0 yo:o:(x) ". . z 0 = a(x)z , /#0 y::(x) { % (21.21). ,#"$, /yo:: = yo:o: + y::(21:24) yo:: { ". (21.21).J' / (21.23) ' $'. " , $( 2!(. ! 21.1. ' ( //0' (21.21). H" ( ".
..2181) @( ". . $ %:dz = a(x)z 6 dz = a(x)dx , Z dz = Z a(x)dx ,dxzzZR a(x)dxR a(x)dx, ln jz j = a(x)dx+ln C1 , jz j = C1e, z = Ce(C = C1):2) ". (21.21) R a(x)dx y = C (x)e(.. # 3 , . , # "" /#0( R x, .. '). /#0 a(x)dxy = C (x)e (21.21), " 'RRRC 0(x)e a(x)dx + C (x)e a(x)dx a(x) = a(x)C (x)e a(x)dx + b(x):R a(x)dx&#. $' C (x)e a(x) (, /#0 () 3' R a(x)dx0;C (x) = eb(x) # % C (x):Z RGC (x) = e; a(x)dxb(x)dx + C:R a(x)dx3) ( /#0 (x) y = C (x)e".
(21.21):Z RRy = yo:: = e a(x)dx CG + e; a(x)dxb(x)dx : 21.8. ' dy = y + x(21:25)dx ( ' # , %. $ # M (16 2):& . # :dz = z 6 dz = dx , Z dz = Z dx , ln jz j = x + ln C ,1dxzz, jz j = C1ex , z = Cex (C = C1):219 (21.25) y = C (x)ex (x) { # $ /#0. y = C (x)ex (21.25):C 0(x)ex + C (x)ex = C (x)ex + x , C 0(x)ex = x , C 0 (x) = xe;x ,ZG, C (x) = xe;xdx , C (x) = ;xe;x ; e;x + C:&', ". (21.25) G x ; (x + 1):y = C (x)ex = (;xe;x ; e;x + CG )ex = Ce3 $' = 1 y = 2, 2 = CG e ; 2 , CG = 4e;1:J, ' #, %. $ # M (16 2) y = 4ex;1 ; (x + 1):, .,# $ y0 = a(x)y + b(x)yn (21:26) # a(x) b(x) { $ $# Ca6 b] /#0, n { (n 6= 0 n 6= 1). ? 3 # ( .
"$:) $ " (21.26) yn, y;n y0 = a(x)y1;n + b(x)6") $ ( z = y1;n , z 0 = (1 ; n)y;n y0 " '1 z 0 = a(x)z + b(x) , z 0 = (1 ; n)a(x)z + (1 ; n)b(x):1;n220 ( , # ". 21.9. ' y0 + 2y = exy2:(21:27)2 " .1. $ " (21.27) y2 $y;1 = z: z 0 = 2z ; ex. (, o 3 ' #$ ". I ' z = e2x C + ex $ % . ". :22xx ;14 y = (e C + e ) y=0( y = 0 " $' y).2. I #' (21.27) $% /#0( y = uv u(x) v(x): $ (21.27), u0 v + uv0 + 2uv = exu2v2 , u0 v + u(v0 + 2v) = exu2v2:", " v0 + 2v = 0: , v = e;2x " u0 = e;xu2: $ %, (, du = e;xdx , Z du = Z e;xdx , ; 1 = ;e;x ; C , u = (e;x + C );1:u2u2uJ y = uv = e;2x(e;x + C );1 = (Ce2x + ex);1: 2 y = 0 # v:, ? M (x y)dx + N (x y)dy = 0221(21:28) ' //0 #( /#0u(x y) ..
du(x y) M (x y)dx + N (x y)dy $ % //0%. 8 u(x y) = C { ".( . 21.2. * M (x y) N (x y) @M=@y @N=@x D. /! (21.28) ! !, !@N (x y) @M (x y) (8(x y) 2 D):(21:29)@x@y4#$' ( # # , (21.29) "% ./#0 u(x y) #(, du = Mdx + Ndy: " #$' /# #( #.? (21.29), /#0 u = u(x y)$ (@u = M (x y)@u = N (x y):@x@y 21.10.
' ( + y + 1)dx + (x ; y2 + 3)dy = 0:(21:30) (21.29):@ (x ; y2 + 3) 1 @ (x + y + 1) 1:@x@y % x y, $ (21.30) { %//0%. &', . /#0 u(x y) #,@u = x + y + 1 @u = x ; y2 + 3:(21:31)@x@y , $ "% ( ( . I ' u = x2=2 + xy + x + '(y) ($' '222$'( ( /#0 '(y) $.
). /#0 (21.31), " '3y0202x + ' (y) = x ; y + 3 , ' (y) = 3 ; y , '(y) = 3y ; 3 + C1:&', u = x2=2 + xy + x + 3y ; y3 =3 + C1 ".( (21.30) u(x y) = C2: #'( :x2 + xy + x + 3y ; y3 = C:238 #% %, # (21.28) % //0%, 3 ( # /#0 (x y) 3 # "% ( (21.28) ( M )dx + ( N )dy = 0 % //0%. 8 /#0 (x y) $ ! (. V, (x y) " . 3 (21.28), (x y) % $%@ (M ) = @ (N )@y@x# % ## 3 ' @N @M !@@(21:32)M @y ; N @x = @x ; @y :' ". $( '(. # 3 $' 0' #' .(3' /#0 ( ( /#0, $.( '# ($) x y x2 + y2 x y ..='$ (21.32), #$', (x y) (x) /#0' = (@M=@y ; @N=@x)=N $ '# x. 8 R '(x)dx = (x) = e223:(21:33)? 3 /#0 ' = (@M=@y ; @N=@x)=(xN ; yM ) $ '# x2 + y2 .( 3' R221=2 '(z )dz = (x + y ) = e:z =x2 +y2=, #0, /#0 @N @M ! @!!@!' = @x ; @y = M @y ; N @x(21:34)$ '# ! = !(x y) (21.28) # .( 3' = (!(x y)) ( /R '(!)d! :!=!(xy)=e(21:35)4(', ' /#0 (21.34) $ '# !: (21.35) (21.32), R '(!)d!R '(!)d! @N @M !R '(!)d!@!@! '(!) @x e @x ; @y :Me '(!) @y ; Ne&#.
$' # o '(!) /#0 (21.34), 3, $ ., /#0 (21.35) (21.32). &', (21.35) { .( 3' (21.28). 21.11. @( ".( (1 ; xe;y )dx + (x + 2xe2y + x2e;y )dy = 0:(21:36)=@M = xe;y @N = 1 + 2e2y + 2xe;y :@y@x,# ## @M=@y 6= @N=@x, (21.36) %//0%. " ( .( 3'. ,### /#0;y ; 1 ; 2e2y ; 2xe;yxe1@M=@y;@N=@x'===;Nx(1 + 2e2y + xe;y )x224$ '# x (21.36) .( 3'RR = (x) = e '(x)dx = e; dx=x = x1 (x 6= 0):A3 " (21.36) 1=x !1;y2y;y x)dy = 0;edx+(1+2e+ex % //0%.
@% /#0 u = u(x y) $ (@u = 1 ; e;y @u = 1 + 2e2y + e;y x:@x x@y=$ u = ln jxj;e;y x+v(y) , , e;y x + v0 (y) = 1 + 2e2y + e;y x , v0(y) = 1 + 2e2y , v(y) = y + e2y + C1:&', u(x y) = ln jxj; xe;y + y + e2y : #'( :22y;y4 ln jxj ; xe + y + e = Cx = 0: 21.12. ' xdx + ydy + xdy ; ydx = 0:(21:37)J' /#0; @N=@x = (@=@y)(x ; y) ; (@=@x)(x + y) = ; 2' = @M=@yxN ; yMx(x + y) ; y(x ; y)x2 + y2$ '# ! = x2 + y2 : J, /#0RR dw=w 1221=2 '(!)d!; = (x + y ) = e=e= w = x2 +1 y2 . 3 (21.37).
A3 " = 1=(x2 + y2 ), xdx + ydy + xdy ; ydx = 0 , 1 d(ln(x2 + y2)) + d(y=x) = 0:x2 + y2x2 + y221 + (y=x)2225p= , ln x2 + y2 + arctg(y=x) = ln C: "$( :qx2 + y2 earctg(y=x) = C:,, !dyF x y dx = 0:(21:38)? 3 # $' ' dy=dx, #'# ( dy = f (x y) i = 1 mdx i# $3 ".
# $' (21.38) ' $( (' " ' ). , (21.38) # $( , # (% .) A (21.38) 3 $ /#0 y = y(x).. F (x y0 ) = 0:(21:39)" /#0 '(t) (t) #, " ' 3 F ('(t) (t)) 0: , x = '(t) y0 = yx0 = (t): = '# y = y(x) (21.39) " $ #, c x = '(t) ( # y. 4 '$ /( yx0 = yt0 =x0t , yt0 = yx0 x0t:I '0yt = (t) '0(t) , y = Z (t)'0(t)dt:, # Zx = '(t) y = (t)'0 (t)dt (21.39).226 21.13. ' qx 1 + y02 = y0 :(21:40)J' 0"$ t #, " y0 = yx0 = tg t: , t 2 (;=26 =2) $ (21.40) x = tg t , x = sin t:cos t,# "$, ( /#0 x = sin t y0 = tg t . (21.40). # ( # y.
=yt0 = yx0 xt , yt0 = tg t cos t , yt0 = sin t , y = ; cos t + C:=#, (21.40) #( / x = sin t y = ; cos t + C: ? # ' $' C , x2 +(y ; C )2 = 1 (21.41) "( /.:( x = sin ty = ; cos t + C (t 2 (;=26 =2)) x2 + (y ; C )2 = 1:") A (21.38) 3 $ x, .. F (y y0 ) = 0:(21:41)" /#0 y = '(t) yx0 = (t) #, " '3 F ('(t) (t)) 0: , /#0 y " $, ( # x. =0 (t)0(t)0Z '0 (t)''yt00yx = x0 , (t) = x0 (t) , xt = (t) , x = (t) dt + C 6t # Z 0x = '((tt)) dt + C y = '(t) p (21.41).227 21.14. ' p1 +y y02 = 1:(21:42)H" $"' 0', " t#, " y0 = yx0 = sh t: , y= ch t = 1 $ y = ch t: @(# x.
=0yx = yt0 , sh t = sh t , x0t = 1 , x = Z dt = t + C:x0tx0t&', (21.42) #( / x = t + C y = ch t (;1 < t < +1). =# $' t, 3$' ".( p y = ch(x ; C ) (21.42) "(/. :( x = t + Cy = ch t (t 2 (;16 +1)) y = ch(x ; C ):) A (21.38) # $ ' y, .. # y = f (x y0 ):(21:43)8 # " $' y0 = yx0 = p ($' ' 0, "$ "#( t, "#(p). , (21.43) y = f (x p):2 '# , $ . y x $ p. @ ( . # .//0 y = f (x p) @f dp6 dy = @f + @f dp , p = @f + @f dp :dy = @fdx+@x@pdx @x @p dx@x @p dx //0' ' $( /#0 p = p(x), # ## $ @f=@x @f=@p $ x p.
= //0' , ( ".( W(x p C ) = 0:228, ( W( C ) = 0 y = f (x p), p { , ( '% #% (21.43) #( /.) A (21.38) # $ ' x, .. # x = g(y y0):(21:44)8 #3 " y0 = yx0 = p: , (21.44) x = g(y p):4//0 , " '@g dp6 dx = @g + @g dp 6 1 = @g + @g dp :dx = @gdy+@y@pdy @y @p dy p @y @p dy //0' ' /#0 p = p(y), # ## $@g=@y @g=@p /#0 y p. = //0' , ".( W(y p C ) = 0:, ( W(y C ) = 0 x = g(y p), p { , ( '% #% (21.44) #( /. 21.15. ' By = xy0 + (y0 )(21:45) (u) { # /#0 ( " D.J' (21.45) 3 (21.43).
p = dy=dx y = xp + (p):(21:46)4 , //0 (21.46). 8 :dy = pdx + xdp + 0 (p)dp , dy = pdx + (x + 0(p))dp:,# ## x = C (21.45), dx 6= 0 $ 3 $' dx 'dy = p +(x + 0 (p)) dp , p = p +(x + 0 (p)) dp , (x + 0 (p)) dp = 0 ,dxdxdxdx229204 x + (p) = 0, dp=dx = 06204 x + (p) = 0, p = C = const :A (21.46), , ' # (21.45)$ 2 ( y = xp + (p)66 x + 0 (p) = 0664(21:47)y = xC + (C ):F ' .
21.16. ' E 3y = x'(y0 ) + (y0 )(21:48) '(u) 6 u (u) '(u) { # % "% /#0.J' #3 (21.48) (21.43). p = dy=dx y = x'(p) + (p):(21:49)H" ( . , //0 (21.49).8 $' dy = '(p)dx + x'0 (p)dp + 0 (p)dp , dy = '(p)dx + Cx'0 (p) + 0 (p)]dp:#'# = (21.48), , dx, # dy = '(p) + Cx'0 (p) + 0 (p)] dp , p ; '(p) = Cx'0(p) + 0 (p)] dp : (21:50)dxdxdx? dp=dx 6= 0 (21.50) ' .:0 (p) + 0(p):Cp ; '(p)] dx=x'dp ( x, , #'# '(p) 6 p 3 ( ".( W(x p C ) = 0 '$', , 0 $'( c( E 3.? dp=dx = 0 (21.50) # '(p) = p: ? "# ('230# p = pi (i = 1 m) ' # E 3(21.48) (y = x'(p) + (p) , y = x'(p ) + (p ) i = 1 miip = pi6..
. "Z " , , #'( $':2 ( y = x'(p) + (p)66 W(x p C ) = 0664y = x'(pi ) + (pi ) i = 1 m W(x p C ) = 0 { ".( 0 (p) + 0(p)6Cp ; '(p)] dx=x'dpp = pi { # '(p) = p i = 1 m: 21.17. #$', ' #, $ y = xp + (p) x = ;0 (p) " .( ( '% #% y = xC + (C ) B (21.45).@, " . ( '% #% ( 3 '%) W(x y C ) = 0 % $ ((W(x y C ) = 0(21:51)@ W(x y C )=@C = 0 # C .
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