1. Интегралы ФНП Диф_ур (853736), страница 33

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 33)

5 y00 + 2y0 + 5y = e;x cos 2x:< ) Y 2 + 2 + 5 = 0 1:pp12 = ;1 1 ; 5 = ;1 ;4 = ;1 2i5199yoo = C1e;x cos 2x + C2e;x sin 2x:) f (x) = e;x cos 2x  f (x) = e;1x(1 cos 2x + 0 sin 2x):F + i = ;1 + 2i , 9 y (23.4) = ;1 = 2 k = 0 ..y = xe;x(A cos 2x + B sin 2x)0 = e;xN(A ; Ax + 2Bx) cos 2x + (B ; Bx ; 2Ax) sin 2x]y00 = e;x N(;2A ; 3Ax + 4B ; 4Bx) cos 2x+y+(;2B ; 3Bx ; 4A + 4Ax) sin 2x]:0 y 00 y y;xe 6= 0:;4A sin 2x + 4B cos 2x = cos 2x: 9 , :9cos 2x 4B = 1 = sin 2x ;4A = 0 o A = 0, B = 14 . 2,y = xe;x 14 sin 2x:) -  y = yoo + y:y = e;x(C1 cos 2x + C2 sin 2x) + 41 xe;x sin 2x: > 23.6.

5 y00 + 10y0 + 25y = 4 ch 5x:520< ) Y 2 + 10 + 25 = 0 ( + 5)2 = 0:. 1 = ;5 2, 9yoo = C1e;5x + C2xe;5x:5x + e;5xe ) = ch 5x =2 f (x) = 2e5x + 2e;5x:(23:6) 4 y  : y1 { f1(x) y2 { f2(x) y = y1 + y2 { f (x) = f1(x) + f2(x): (23.6) f1(x) = 2e5x, f2(x) = 2e;5x. = 5 , y1 (23:40)y1 = Ae5x ;5 , y2 (23:50)y2 = x2e;5xB:= , y y1 + y2, ..y = Ae5x + Bx2 e;5x0 = 5Ae5x + B (2x ; 5x2 )e;5x y00 = 25Ae5x + B (2 ; 20x + 25x2 )e;5x:y0 y 00 y yy00 + 10y0 + 25y = 2e5x + 2e;5x100Ae5x + 2Be;5x = 2e5x + 2e;5x:521 9 e5x e;5x:9e5x 100A = 2 = e;5x 2B = 2 1 , B = 1, c, A = 501 e5x + x2e;5x:y = 50) -  y = yoo + y:1 e5x + x2e;5x: >y = C1e;5x + C2xe;5x + 50 23.7.

J  ' 8 00< y ; y = 4 cos x: y(0) = 0 y0 (0) = 1:< ) Y 2 ; 1 = 0 ( ; 1)( + 1) = 0 1 = 1 2 = ;1 9yoo = C1ex + C2e;x:) 4 cos x  f (x) = e0x(4 cos x + 0 sin x):F + i = 0 + 1i = i , 9 y (23.4) = 0 = 1 k = 0y = A cos x + B sin x0 = ;A sin x + B cos xy00 = ;A cos x ; B sin x:y5220 y 00 : y y;A cos x ; B sin x ; A cos x ; B sin x = 4 cos x;2A cos x ; 2B sin x = 4 cos x A = ;2, B = 0, c,y = ;2 cos x:) - y = yoo + yy = C1ex + C2e;x ; 2 cos x:(23:7)4 C1 C2  , y0 = C1ex ; C2e;x + 2 sin x:7y(0) = C1e0 + C2e;0 ; 2 cos 0 = 0y0 (0) = C1e0 ; C2e;0 + 2 sin 0 = 1: (C + C ; 2 = 012C1 ; C2 = 1 C1 = 32 C2 = 12 : (23.7) C1 C2:y = 32 ex + 21 e;x ; 2 cos x: > &,$ .1) y 00 2y 0 3y = e4x; ;3) y ; 2y + y = 6xex5) y ; 4y + 8y = e2x + sin 2x000000y + y = 4xex4) y + y = x sin x6) y ; 5y = 3x2 + sin 5x2)0000005230y + 2y + 2y = xe x y(0) = 0 y (0) = 08) y ; 2y = 2ex y (1) = ;1 y (1) = 09) y ; 3y ; 2y = 9e2x y (0) = 0 y (0) = ;3 y (0) = 310) y IV + y = 2 cos x y (0) = ;2 y (0) = 1 y (0) = 0 y7)000000000;00000000000(0) = 0:000y = C1e x + C2e3x + 15 e4x2) y = C1 cos x + C2 sin x + (2x ; 2)ex 3) y = (C1 + C2 x + x3)ex2!xx4) y = C ;cos x + C +sin x1);1244y = e2x (C1 cos 2x + C2 sin 2x) + 0 25e2x + 0 1 cos 2x + 0 05 sin 2x6) y = C1 + C2 e5x ; 0 2x3 ; 0 12x2 ; 0 048x + 0 02(cos 5x ; sin 5x)7) y = e x (x ; sin x)8) y = e2x 1 ; 2ex + e ; 19) y = (x ; 1)(e2x ; e x )10) y = x ; x sin x ; 2 cos x:5);;;||||{524 24 J , ..

dxk = f (t x x ::: x ) k = 1 2 ::: n k1 2ndt xk = xk (t) { , fk {  n + 1 .( *2  . 24.1. J 8>dx>< dt = y + 1>dy>: = x + 1:dt< 7 y = dxdt ; 1 dy = d dx ; 1! = d2x :dt dt dtdt25  dy dt d2x = x + 1 d2x ; x = 1:(24:1)dt2dt2 { 9 (L54A) 2- .

(S L54A  U22 U23.) 5 (24.1).) Y 2 ; 1 = 0 ( + 1)( ; 1) = 0:525(24:2). 1 = ;1 2 = 1 1, 9xoo = C1e;1t + C2e1t:) (24.1) f (t) = 1 f (t) = e0t 1:F = 0 (24.2), 9x = A ) x00 = 0 , x x00 (24.1), 0 ; A = 1 A = ;1 , , x = ;1:) x = x = x + x 9;tt=;Cx = C1e;t + C2et ; 1 dx1 e + C2 e dt.. y = dxdt ; 1 ;tty = dx;1=;C1 e + C2 e ; 1:dt= , :x = C1e;t + C2et ; 1 y = ;C1e;t + C2et ; 1: > 24.2. J  ' 8>dx>< dt = 3x + 8y>dy>x(0) = 6 y(0) = ;2:: = ;x ; 3ydt< 7 x = ;3y ; dydt dx = ;3 dy ; d2y :(24:3)dtdt dt2 (24.3) , 2y2ydydddy;3 dt ; dt2 = ;9y ; 3 dt + 8y dt2 ; y = 0526 9 L-4A. .

2 ; 1 = 0 ( ; 1)( + 1) = 0: 2,t;ty = C1et + C2e;t dy=C1 e ; C2e dt x = ;3y ; dydt x = ;4C1et ; 2C2e;t:- x = ;4C1et ; 2C2e;t y = C1et + C2e;t:(24:4)4 C1 C2  9x(0) = 6 = ;4C1 ; 2C2 = y(0) = ;2 = C1 + C2 C1 = ;1 C2 = ;1 (24.4) :x = 4et + 2e;t y = ;et ; e;t : > &,$ #' .8 dx>< dt = ;9y1)>: dy = xdt8 dx>>< dt + 3x + 4y = 0dy + 2x + 5y = 03)>>: xdt(0) = 1 y(0) = 48 dx>>>dt = ;y + z< dy5)>dt = z>>: dz = ;x + z:dt2)4)5278 dx>< dt = y + t>: dy = x ; tdt8 dx dy>< 4 dt ; dt + 3x = sin t>: dx + y = cos tdtx = 3C1 cos 3t ; 3C2 sin 3t y = C1 sin 3t + C2 cos 3t2) x = C1et ; C2e t + t ; 1 y = C1et + C2e t ; t + 13) x = ;2e t + 3e 7t y = e t + 3e 7t 4) x = C1e t + C2 e 3t y = C1 e t + 3C2 e 3t + cos t8 x = (C1 ; C2) cos t + (C1 + C2) sin t<5) y = C1 sin t ; C2 cos t + C2et :z = C1 cos t + C2 sin t + C3et:1);;;;;;;;;;||||{( 2   , .

24.3. J 8>dx>< dt = x2 + y2>dy>: = 2xy:dt< 2 , dx + dy = x2 + y2 + 2xy d(x + y) = (x + y)2 : (24:5)dt dtdt- x + y = u du = u2 du = dtdtu2Z du Z1 = t + C u = ; 1 x + y = ; 1 : (24:6)=dt;1u2ut + C1t + C1:  , d(x ; y) = (x ; y)2(24:7)dt5289 (24.5) x ; y = ; t +1C :2: (24.6) x y: 81>>< x + y = ; t + C1 >1>: x ; y = ;t + C :22 ,  , !11(24:8)2x = ; t + C + t + C 2y = t +1C ; t +1C :1221F,  (24.6) (24.7) x + y = 0 x ; y = 0 ..

x 0 y 0: : (24.8) !!111111x = ; 2 t + C + t + C y = 2 t + C ; t + C 1221x = 0 y = 0: > 24.4. J 8 dx1>>< dt = ; y >dy 1>: = :dt x< A y, { x :dy = 0:+xy dxdt dt dy = d(xy)+xy dxdt dt.. d(xy) = 0 xy = C1 y = Cx1 :(24:9)529 (24.9) :dy = d C1 ! = d C1 ! dx = ; C1 dx ..dt dt xdx x dtx2 dtdy = ; C1 dx : dy , dt x2 dt dt:1 dx = ; 1 dt; Cx21 dx=dt xxC1Z dxZ11 t + ln jC j=;dtlnjxj=;2xC1C1 ln Cx = ; C1 t Cx = e;(1=C )t x = C2e;t=C :212115 x (24.9):y = C eC;1t=C y = CC1 et=C :221 t=C x = C2e;t=C y = CC2 e : > 24.5.

J  ' 8 dx1>>=1;< dty>1 x(0) = 1 y(0) = 1:dy>: =dt x ; t< A y8, { 01 (x;t) 8!>d(x ; t)>dx ; 1 = ;1>>>y< y @ dt A = ;1< dt >>dydy = 1:>>>: (x ; t) = 1(x;t):dtdt2 :d (y(x ; t)) = 0 y d(xdt; t) + (x ; t) dy=0dtdty(x ; t) = C1 x ; t = Cy1 :5301111 C1=y x ; t  dy = y dy = 1 dtdt C1y C1 Z dyZ11 y 1 t y = C et=C :=dtlnjyj=t+lnjC2j ln =2y C1C1C2 C1(7,(x ; t)y = C1y = C2et=C x = t + CC1 e;t=C y = C2et=C :2 t = 0,  1 = C1=C2, 1 = C2, ..C1 = C2 = 1 x = t + e;t y = et: >1111 &,$ +0 !7.8 dx>< = sin x cos y1) > dt: dy = cos x sin ydt8 dxy>>< dt = x ; y 3)dyx>>: dt = x ; y 8 dx 1>< et dt = y 2) >: et dy = 1 dt x8 dx x>< dt = y 4)>: dx = y dt x1) tgx + y = C et21tgx ; y = C et22e;t 2y = C1x C1x2 = C2 ;3) x2 ; y 2 = C1 x ; y + t = C2 114) ; = C1 1 + C1x = C2eC1 t :x y2)||||{531( 02 9  .A.

. . 24.6. J 8>dx = 3x ; y + z>>>dt>< dy(24:10)= ;x + 5y ; z>dt>>dz>>: = x ; y + 3z:dt< ) : 103;11A = BB@ ;1 5 ;1 CCA1 ;1 3 c det(A ; E ) = 0  1 3 ; ;1 ;1 5 ; ;1 = 0:1;1 3 ; J , , , 3 ; 112 + 36 ; 36 = 0:(24:11). (& A) 1 = 2 2 = 33 = 6 1. (F, , , 1 = 2 , ;36 (24.11),  (24.11) ( ; 2):)532) 4! = f1 2 3g A  (A ; E ) !=!o :(24:12)1: = 1 = 2:: (24.12) = 2  010 1 0 13;2;11BBCC BB 1 CC BB 0 CC@ ;1 5 ; 2 ;1 A @ 2 A = @ 0 A 1;1 3 ; 2 308>< 1 ; 2 + 3 = 0;1 + 32 ; 3 = 0(24:13)>: 1 ; 2 + 3 = 0:: (24.13) , 9 (1 ; 2 + 3 = 0;1 + 32 ; 3 = 0: , (1 ; 2 + 3 = 022 = 02 = 0 1 = ;3: :, , 1 = 1 3 = ;1 A! = f1@ 0@ ;1g:5 (24.10) ! t e 0 1 0 1BB x CC BB 1 CC 2t(24:14)@y A = @ 0 Ae :z;12: = 2 = 3: : (24.12) = 3  10 1 0 100;11CC BB 1 CC BB 0 CCBB@ ;1 2 ;1 A @ 2 A = @ 0 A 01 ;1 0 353318>< ;2 + 3 = 0;1 + 22 ; 3 = 0>: 1 ; 2 = 0 :7  1 = 2 = 3, 1 = 1 2 = 3 = 1 A! = f1@ 1@ 1g:5 ! e t0 1 0 1BB x CC BB 1 CC 3t(24:15)@y A = @1Ae :z13: = 3 = 6: : (24.12) = 6:010 1 0 1;3;11BBCC BB 1 CC BB 0 CC@ ;1 ;1 ;1 A @ 2 A = @ 0 A 1 ;1 ;3 308>< ;31 ; 2 + 3 = 0;1 ; 2 ; 3 = 0>: 1 ; 2 ; 33 = 0:2 , ;22 ; 43 = 0, 3 = 1 2 = ;2,  1 = 1: = A! = f1@ ;2@ 1g (24.10) ! e t 0 1 0 1BB x CC BB 1 CC 6t(24:16)@ y A = @ ;2 A e :z1) -   (24.14), (24.15), (24.16)0 10 10 10 11x1BB 1 CC 6tCC 2tBB CC 3tBB CBCB@ y A = C1 @ 0 A e + C2 @ 1 A e + C3 @ ;2 A e 11z;15342382t3t6t>< x = C1e3t + C2e 6t+ C3ey = C e ; 2C e>: x = ;2C e2t + C3 e3t + C e6t:>B.

Характеристики

Список файлов лекций