1. Интегралы ФНП Диф_ур (853736), страница 16
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% $ y = C5x5 + C3x3 + C2x2 + C1x + C0:,, &( F (y y0 ::: y(n)) = 0:(22:10)8 $ x: & y $(dy ( y = y(x)). ='$(, /#0 p = p(y) = dx //0 3( /#0, " 'd (y0 ) = d (p(y)) = dp dy = p dp 6y00 = dxdxdy dx dy dp ! d dp ! dy dp !22dddpy000 = dx (y00 ) = dx p dy = dy p dy dx = p2 dy2 + p dy .. , k{ $ /#0 y = y(x) 3 $ /#0 p = p(y) $ k ; 1{ #,245 $ p = dy=dx % (22.10) # F1 (y p p0 ::: p(n;1)) = 0(22:11)# #o 0 ' # (22.10). A (22.11) //0' (n ; 1){ # ' $( /#0 p = p(y): ? ' ". p = '(y C1 ::: Cn;1) # ". % (22.10) " dy = '(x C ::: C ):' dx1n;1 22.6.
@( $ B1 + y02 = 2yy00 y(1) = y0 (1) = 1:(22:12)F 3 $ x 3 # .' $ p = p(y) = dy=dx: =d (y0 ) = d (p(y)) = dp dy = p dp :y00 = dxdxdy dx dy ' (22.12) % dyp(y) jx=1 = dx , p(y(1)) = y0 (1) , p(1) = 1:x=1=#, /#0 p = p(y) ' $ Kodp p(1) = 1:1 + p2 = 2yp dy$ , pdp = 1 dy , Z pdp = 1 Z dy , 1 ln(1+p2) = 1 ln jyj+ 1 ln C ,1 + p2 2y1 + p2 2 y222 1q, 1 + p2 = C1jyj , 1 + p2 = Cy (C = C1) , p = Cy ; 1:pA, p(1) = 1 > 0 3 $' p = + Cy ;p1: $' y = 1 '$' p(1) = 1 " ' 1 = C ; 1 ,pC = 2: &', p = 2y ; 1 '( #:dy = q2y ; 1 , p dy = dx , Z p dy = Z dx ,dx2y ; 12y ; 1246,q2y ; 1 = x + C2:H" ( C2, 3p x = 1:#'# y(1) = 1 " ' 2 1 ; 1 = 1 + C2 , C2 = 0:=#, $ B (22.12) $ q2x2y ; 1 = x , y = 2+ 1 = 21 (1 + x2):,, (22.1).
? ' " (F (x ty ty0 ::: ty(n)) tkF (x y y0 ::: y(n)) # R zdx(22.1) 3 "' 3 0 .' $ y = e z = z (x) { $ /#0.#3 F (x y y0 y00 y000 ) = 0(22:13)R' #. & $ y = e zdx RRRRy0 = e zdxz 6 y00 = e zdxz 2 + e zdxz 0 = e zdx(z 2 + z 0)6RRRy000 = e zdxz (z 2 + z 0) + e zdx(2zz 0 + z 00) = e zdx (z 3 + 3zz 0 + z 00): (22.13), RR zdxR zdx 2 0 R zdx 3zdx000F x e e z e (z + z ) e (z + 3zz + z ) = 0:='$/#0 F , 3 $ $# F R zdx'k3' e: &# " , " 'F (x 1 z z 2 + z 0 z 3 + 3zz 0 + z 00) = 0:2 p # ' $(/#0 z = z (x): ? ( z = '(x C1 C2) 247, ".
% (22.13) R '(xCC )dx : / y = e 22.7. @( ". yy00 ; (y0 )2 = (x + 1)y2:12 yy00 ; y02 ; (x + 1)y2 = 0"3, ' R zdx000k = 2 'y y y : 4 $ y = e : , RRy0 = e zdxz y00 = e zdx(z 2 + z 0) % RRR02 zdx 22 zdx2 zdx 2z ; (x + 1)e= 0:e(z + z ) ; e #. # z 0 = x +1 ". # /#0 z = x2=2+ x + C1 : &',". % " /#0R zdxy=e= ex =6+x =2+C x+C :3212 .
#'# , #% 3 # $ 0, ( .= # "3. 22.8. ' yy0 + x(y0 )2 + xyy00 = 0:F ' y, y0 y00 3 " " ' # (, % ' $( /#0 % $%.# # $', ' ($( /#0 xyy0 : J, (xyy0 )0 = 0 . "$:Z 2C 1C1C11002 02xyy = C 16 yy = x , 2 (y ) = x , y = x dx ,q2, y = 2C 1 ln jxj + C2 , y = 2C 1 ln jxj + C2:248 22.9. #$', /#0 y = y(x) $ -# x = 12 ln t + 43t2 y = 41 t + 43t3 (t > 0)(22:14) (y00 )2 ; 2y0 y00 + 3 = 0:@ ( yx0 yx00 : '$' /( $( /#0,$( #, " ':0yx = y0 = yt0 = 1=4 ; 9=(4t4) = t2 + 3 6x0t 1=(2t) ; 3=(2t3)2t0 )0t22(y(t;3)=(2tx0000yx = y = x0 = 1=(2t) ; 3=(2t)3) = t:t yx0 yx00 y002 ; 2y0 y00 + 3 = 0 32t2t ; 2 2+t 3 t + 3 0:F $, /#0, $ # (22.14), //0' .
22.10. ' $ Bx2yy00 ; (y ; xy0 )2 = 0 y(1) = y0 (1) = 1:(22:15)222#'# ' (22.15) { (R zdx R k = 2, R $ y = e : ,# ##y0 = e zdx z y00 = e zdx (z 2 + z 0) #$( $ " x2(z 2 + z 0) ; (1 ; xz )2 = 0 , x2z 0 ; 1 + 2xz = 06 z 0 = ; x2 z + x12 :F ( //0' ' z . 0 $'( ( E 3 (. #0 21), . : z = Cx21 + x1 : J, ". % (22.15) " ( /R (C =x +1=x)dxR zdx= e;C =x+ln jxj+ln C = C2xe;C =x (C2 = C 2):y=e =e12122491 ' y(1) = y0 (1) = 1 C2e;C = 1 C2 e;C (1 + C1) = 1 , C2e;C = 1 1 + C1 = 1:1110 % C1 = 0 C2 = 1: &', $B (22.15) /#0 y = x: 22.11. =$ ".
//0' y02 + 2yy00 = 0' ' # , %. $ # (1 1) # . ( y = x:' # ' # y = y(x): , #'# % $ # M (1 1), 3 "' y(1) = 1:4, # y = y(x) # ( y = x # M , $ ( #//0 ( # #//0 (y = x .. y0 (1) = 1. =#, ' $ By02 + 2yy00 = 0 y(1) = y0 (1) = 1:,# ## $( ( , 3 # .' $ p = dy=dx. #'#d (y0 ) = d (p(y)) = dp dy = p dp y00 = dxdxdy dx dy #$( $ " dp = 0 ,p2 + 2py dy266 p = 0 , y = C:642y dp = ; p:dy $ %:dp = ; dy , Z dp = ; 1 Z dy , p = p 1 :p2yp2 yC1y@( C1: #'# p(y) = dy=dx x = 1 " 'pC ydy p(y(1)) = dx,p(1)=1:y=1p=1=1x=1250p,C1 = 1. J, p = 1= y ' y0 = 1=py: $ %:dy = 1 , pydy = dx , Z pydy = Z dx , y3=2 = 3 (x + C ):2dx py2 $' x = 1 '$' y(1) = 1 1 = 32 (1 + C2) , C2 = ; 13 : &', % $ B "3 !#2=31 .
: y = 2 x ; 3 : 1. "! 3" "< !?2. ! ! "! < ! n{ !?3. ! $3! $ ($ <) ! ! ! "<?4. ! <! $ Ko<! ! n{ !. ?$ 1 3" &? =# $ < 3 <, "!! &! 9" 1 "?5. ! ! "! 9 < ! n{ !? @ $ < 1 !?6. !! 9 ! "< !? @ $" ?7. <! y(n) = f (x)?8. 2< #! ! ! ! F (x y(k) ::: y(n)) = 0:9. #! ! 3 !, ! ! x?10. "! " 3 & "&? #! ! & ?251 23-( '** ,( + !!(0" '!. *+' 0- + + ;<= 0 ' '= + !.' ' + !, P+ + =;<= 0 !(/ / = 0 ' = )2 n{ # $pn(x)y(n) + pn;1(x)y(n;1) + ::: + p1(x)y0 + p0(x)y = f (x)(23:1) # $ /#0 y = y(x) $ %( "$ (..
0( 0'( ' (). /#0 pn(x) ::: p0(x) $ (23.1), ' f (x) { . ? (23.1) ' (f (x) 0) (23.1) $ . ? 3 f (x) 6 0 (23.1) $ //0' .A (23.1) 3 $' ## Ly = f (x), "$'$ L { //0'( n{ o #:n;1ndd + p (x):d(23:20)L = pn(x) dxn + pn;1 dxn;1 + ::: + p1(x) dx02 , ' $# ( (. 8 '( " '$' . :1) C Ca6 b] { /#0(, % $# Ca6 b]62) C k Ca6 b] { /#0( y = y(x) % $ y0 (x) ::: y(k)(x) ( k{ # # '),k 1:F ( " /#0( 0 3 3 . 23.1.
(23:20) pn(x) ::: p0(x) Ca6 b], L C nCa6 b] C Ca6 b] (.. L : C nCa6 b] ! C Ca6 b])252 , ..LCC1f (x) + C2g(x)] = C1Lf (x) + C2Lg(x) C1 C2 f (x)g(x) 2 C nCa6 b]:4(', //0 #' /#0 0, $ n{# //0 /#0# C nCa6 b] % /#0 # C Ca6 b]: B , #'# 0 //0 (, L:I ' (23.1) #//0 pn(x) 6= 0 (8x 2 Ca6 b]): B 3' (23.1) $' /Ly y(n) + an;1(x)y(n;1) + ::: + a1(x)y0 + a0(x)y = h(x) "$:ai(x) ppi((xx)) 6 h(x) pf ((xx)) nnnn;1ddd + a (x):L dxn + an;1(x) dxn;1 + ::: + a1(x) dx0@ "3( $ { $' ( ( . @ .$ B 0 2 Ca6 b] (y0 y10 ::: yn0;1) { $' # $ Rn :Ly = h(x) y(x0) = y0 y0 (x0) = y10 ::: y(n;1)(x0) = yn0;1:(23:2) 23.2. (23.2) ai(x) h(x) Ca6 b], $)(23.2) ! ) y = y(x) ) ( .,# "$, . '( $ ( //0' " "'(" %# "#' " %# ".( 20.1 ( .253% .
/. (- ' /#0 y1 (x) ::: yn(x) $# Ca6 b]:#$ 23.1. M, y1(x) ::: yn(x) $# Ca6 b], . 1 ::: n, , #, 3(23:3)1 y1 (x) + ::: + n yn (x) 0 (8x 2 Ca6 b]):? 3 3 (23.3), 1 ::: n { , '# , # i ( 1 = ::: = n = 0) /#0( y1(x) ::: yn(x) $ $# Ca6 b]:; 3#% (a6 b) (a6 b] Ca6 b) # ( "# 3#.J, 31 y1 (x) + ::: +n yn (x) $ (( #"0( /#0(y1(x) ::: yn(x) 1 = 2 = ::: = n { #//0 (( #"0. 23.1.
4#$', /#0(y0 = 1 y1 = x ::: yn = xn(23:4)( $ " $# Ca6 b]:& ( #"0 /#0( (23.4) , # 3 ". ':n0 1 + 1 x + ::: + n x 0 (8x 2 Ca6 b]):& #//0 0 ::: n: & 3$, " x $ $# Ca6 b] # . ? % " $ #//0 i , ' ", #$( " #(, $3. J, i , /#0 (23.4) ( $ $# Ca6 b]: 23.2.
I ( $ 3#(;16 +1) /#0 y1 = sin2 x y2 = cos2 x y3 = 1?254E( #"0 1 sin2 x + 2 cos2 x + 3 1 3 ". ' 3# (;16 +1), $' 1 = 2 = 1 3 = ;1: ,# ## ( " "," % " $ % ' ), #$ /#0( $ 3# (;16 +1): : . 23.3. y1(x) ::: yn(x) c Ca6 b] ! ( ).
3 : y1(x) ::: yn(x) Ca6 b] !, y1(x) ::: yn(x) Ca6 b]:". ' /#0 y1 (x) ::: yn(x) ( $ $# Ca6 b]: , ( 1 ::: n , #, 1y1(x) + ::: + k yk (x) + ::: + nyn (x) 0: ',, k 6= 0: , 3 $'yk (x) ;1k!y1(x) + ::: + ;nk!yn (x) (8x 2 Ca6 b])../#0 k (x) (( #"0( /#0(y1(x) ::: yk;1(x) ::: yk+1(x) ::: yn(x): ": 3yk (x) 1y1(x)+ ::: + k;1 yk;1 (x)+ k+1 yk+1 (x)+ ::: + n yn(x) (8x 2 Ca6 b]) 1y1(x)+ ::: +(;1)yk (x)+ ::: + n yn(x) 0: 2 , 3(23.3) % 1 ::: n % .&', /#0( y1(x) ::: yn(x) ( $. , #$. .
3.10) ? /#0( y1(x) ::: yn(x) 3 /#0 yi (x) 0 ( $ ( $# Ca6 b], # #$ /#0 ).20) ? ##-"' /#0( y1(x) ::: yn(x)( $, y1(x) ::: yn(x) ( $.30) ? /#0( y1(x) ::: yn(x) ( $ $# Ca6 b], ( $ " $# Ca16 b1] 3. $# Ca6 b]:25540) ? /#0( y1(x) ::: yn(x) ( $ $# Ca6 b] ( $ " $# Cc6 d], 3. $# C6 b] (, #, /#0 y1(x) ::: yn(x) $# Cc6 d]).J, ( (( $ /#0( '$ 3' "'( $#, ( (( $ {$' '( $#.4 //#( " # (( $ (( $ /#0( .' 8# .#$ 23.2.
3 9! ( #) /#0( y1(x) ::: yn(x), 3.% C n;1Ca6 b], $ ' y1(x) y0 (x)W (x) W Cy1 ::: yn] = 1 .. y(n;1) (x)1y2(x)y20 (x)..y2(n;1) (x): : : yn(x) : : : yn0 (x) .. ...: : : yn(n;1) (x) # # "$ /#0 y1(x) ::: yn(x) . # $ /#0( .( #. 20 " $' 9!. 23.4 (B4#$ C$ $DD 6$$$EC?F$D). y1(x) ::: yn(x) Ca6 b], ( , .. W Cy1 ::: yn] 0 (8x 2 Ca6 b]):". #'# /#0 y1(x) ::: yn(x) ( $ $# Ca6 b] . 1 ::: n , #, 3y (x) + ::: + nyn(x) 0 (8x 2 Ca6 b]):1 14//0 3 n ; 1 $, .
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