1. Интегралы ФНП Диф_ур (853736), страница 32
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- : y = ; 2x ; 1 >. &,$ .11) y 00 = x3) x4 y 00 + x3y 0 = 1xy(4) = 14) xy + 2y = 06) yy ; y (1 + y ) = 08) (x + 1)y ; (x + 2)y + x + 2 = 02)00000y = 1 ; (y )27) (y )2 + (y )2 = 19) (1 + x2)y ; 2xy = 0 y (0) = 0 y (0) = 310) 1 + (y )2 = 2yy y (1) = 1 y (1) = 111) yy + (y )2 = (y )3 y (0) = 1 y (0) = 11112) y (1 + ln x) + y = 2 + ln x y (1) = y (1) = 1x25)00000000000000000000000000000000; y y(1) = ; 14 y (1) = 12 14) 2y ; 3(y )2 = 0 y (0) = ;3 y (0) = 1 y (0) = ;1:13)y00= (y 0)200000000y = x ln jxj + C1x + C213) y = 2 + C1 ln jxj + C24x2) 6y = x3 ln5)6)1)y = ln je2x + C1j ; x + C2y = sin(C1 + x) + C2x + C39) y = x3 + 3x7)y = x + 1x2 ; 1 13) y =11)42jxj + C1x3 + C2x2 + C3x + C44) y = C1 ln jxj + C2 x + C3 y = C1eC2x + C1 2x8) y = (C1 e + 1)x + C2 110) y = (x2 + 1)12)y21= x2 214) y (x + 2) =||||{505;x ; 6: 22 ) I.
) (L-4A) 9y(n) + an;1y(n;1) + ::: + a1y0 + a0y = 0(22:1)a0 a1 ::: an { .+ ) 4 (22.1) :n + an;1n;1 + ::: + a1 + a0 = 0(22:2) .) 5 (22.2) (?2J) (22.1). 9{ r 1 r (22.1)ex xex ::: xr;1ex@{ - i r 2r (22.1)ex cos x xex cos x ::: xr;1ex cos xex sin x xex sin x ::: xr;1ex sin x:) - y (22.1) { n . 22.1. J y(4) ; 2y000 ; 3y00 = 0:< ) Y 4 ; 23 ; 32 = 02(2 ; 2 ; 3):()506) 5 1 = 0 2 = ;1 3 = 3. 4 () ( ; 0)2( + 1)1( ; 3)1 = 0: . ' 1 = 0 2 y1 = e0x = 1 y2 = xe0x = x:' 2 = ;1 3 = 3 1 y3 = e;x y4 = e3x: = , ?2J, :y1 = 1 y2 = x y3 = e;x y4 = e3x:y = C1 + C2x + C3e;x + C4e3x: > 22.2. J y000 + 4y00 + 4y0 = 0:< ) Y 3 + 42 + 4 = 0(2 + 4 + 4) = 0( + 2)2 = 0:) ' 1 = 0 1 y1 = e0x = 1, 2 = ;2 2 y2 = e;2x y3 = xe;2x: 5 ?2J:y1 = 1 y2 = e;2x y3 = xe;2x:y = C1 + C2e;2x + C3xe;2x: > 22.3.
J y00 + 4y0 + 13y = 0:507< ) Y 2 + 4 + 13 = 0:) 5 pp12 = ;2 4 ; 13 = ;2 ;9 = ;2 3i: - ;2 3i ( 1) y1 = e;2x cos 3x y2 = e;2x sin 3x, ?2J.y = C1e;2x cos 3x + C2e;2x sin 3x: > 22.4. J y(5) ; 2y(4) + 2y000 ; 4y00 + y0 ; 2y = 0:< ) : 5 ; 24 + 23 ; 42 + ; 2 = 0:) 4 (5 ; 24) + (23 ; 42) + ( ; 2) = 04( ; 2) + 22( ; 2) + ( ; 2) = 0( ; 2)(4 + 22 + 1) = 0:= { , ( ; 2)(2 + 1)2 = 0 1 = 2 1 23 = i 2.' 1 = 2 y1 = e2x 0 1i y2 = e0x cos x y4 = xe0x cos x508y3 = e0x sin x y5 = xe0x sin x:y = C1e2x + C2e0x cos x + C3e0x sin x + C4 xe0x cos x + C5 xe0x sin x: >.
= , , n. &,$ .y + y ; 2y = 02) y + 4y + 3y = 03) y + 2y = 04) y ; 5y + 2y = 05) y ; 4y + 5y = 01)000000000000000y7) y6)00+ 2y 0 + 10y = 0+ 4y = 08) 4y + 4y 0 + y = 09) y (5) 10y 000 + 9y 0 = 010) y (5) + 8y 000 + 16y = 0:0000;y = C1ex + C2e 2x2) y = C1 e x + C2e 3x 3) y = C1 + C2 e2x4) y = C1 e2x + C2ex=25) y = e2x(C1 cos x + C2 sin x)6) y = e x (C1 cos 3x + C2 sin 3x)7) y = C1 cos 2 + C2 sin 2x8) y = e x=2 (C1 + C2 x)9) y = C1 + C2 x + C3e x + C4 e3x + C5 e 3x10) y = C1 + (C2 + C3 x) cos 2x + (C4 + C5 x) sin 2x:1);;;;;;;||||{) . ()L (L54A) n- (22:3)y(n) + an;1y(n;1) + ::: + a1y0 + a0y = f (x) f (x) { .
J (22.3) L ( ) .509) 5 ?2J y1(x) y2(x) ::: yn(x) L-4A (22.1)y = C1y1 + C2y2 + ::: + Cnyn:) - (22.3) y = C1(x)y1 + C2(x)y2 + ::: + Cn(x)yn :(22:4)4 C1(x) C2(x) ::: Cn(x) C10 (x) C20 (x) ::: Cn0 (x)8 0>C1y1 + C20 y2 + ::: + Cn0 yn = 0>>>< C10 y10 + C20 y20 + ::: + Cn0 yn0 = 0..>>>>: C10 y1(n;1) + C20 y2(n;1) + ::: + Cn0 yn(n;1) = f (x):) 5 C10 C20 ::: Cn0 { C1(x) C2(x) ::: Cn(x).- (22.4). 22.5. J y000 + y0 = tg x:< ) Y L-4A3 + = 0 (2 + 1) = 0 1 = 0 23 = i , ,yoo = C1 + C2 cos x + C3 sin xy1 = 1 y2 = cos x y3 = sin x ?2J L-4A.) - y = C1(x) + C2(x) cos x + C3(x) sin x:y10 = 10 = 0 y20 = (cos x)0 = ; sin x y30 = (sin x)0 = cos xy100 = 0 y200 = ; cos x y300 = ; sin x 9 C10 (x) C20 (x) C30 (x) 8 00>C30 sin x = 0>< C1 +0 C2 cos x +;C2 sin x + C30 cos x = 0>>: ;C20 cos x ; C30 sin x = tg x:510) J 9 , , :2xsin0001 = tg x C2 = ; sin x C3 = ; cos x :7, Z d(cos x)ZZ sin xdx=;C1 = tg xdx = cosxcos x = ; ln j cos xj + AZC2 = (; sin x)dx = cos x + BZ 2xZ 1 ; cos2 xZ 1ZC3 = ; sin=;dx=;dx+cos xdx =cos xcosxcosx!x= ; ln tg 2 + 4 + sin x + C @ A B C { .
5 C1(x)C2(x) C3(x) ) y = (; ln j cos xj + A) + (cos x + B ) cos x+!!x+ sin x ; ln tg 2 + 4 + C sin x:J A = C1 B = C2 C = C3 : x!y = C1 + C2 cos x + C3 sin x ; ln j cos xj ; sin x ln tg 2 + 4 + 1: > 22.6. J ' L54A8>< y00 + 3y0 + 2y = 1x+1(22:5)e>: y(0) = 2 ln 2 y0 (0) = ;3 ln 2:< ) Y L-4A2 + 3 + 2 = 0 ( + 1)( + 2) = 0 1 = ;1 2 = ;2, 9 y1 = e;x y2 = e;2x ?2J L-4A , ,yoo = C1e;x + C2e;2x:511) - y = C1(x)e;x + C2(x)e;2x:=.. y10 ; e;x y20 = ;2e;2x, C1(x) C2(x) 8 0 ;x>< C1e + C20 e;2x = 01(22:6)>: ;C10 e;x ; 2C20 e;2x = ex + 1 :) J 9 , , :2xxee00C1 = ex + 1 C2 = ; ex + 1 :7, xxZZC1 = exe+ 1 dx = d(eex ++11) = ln(ex + 1) + AZ e2xZ exd(ex)Z (ex + 1) ; 1x) =C2 = ; ex + 1 dx = ; ex + 1 = ;d(exe +1ZZ 1= ex + 1 d(ex) ; 1 d(ex) = ln(ex + 1) ; ex + B @ A B { .
5 C1 C2 ) y = (ln(ex + 1) + A)e;x + (ln(ex + 1) ; ex + B )e;2x :7 (22.6), y0 :y0 = ;e;x (ln(ex + 1) + A) ; 2e;2x(ln(ex + 1) ; ex + B ):A (22.5), y(0) = 2 ln 2 = 2 ln 2 + A + B ; 1 A + B = 1:7 (22.5)y0 (0) = ;3 ln 2 = ;(ln 2 + A) ; 2(ln 2 ; 1 + B )512(22:7) A + 2B = 2:J 8< A+B = 1: A + 2B = 2 , A = 0 B = 1 (22.7) ' y = e;x ln(ex + 1) + e;2x(ln(ex + 1) ; ex + 1): > &,$ .1)e xx1+y =4) y + 4y = 2 tg xsin x+ y = 2 sec3 x6) y + y = ctg x2 y 1 = 1 y 1 = 2 + 2y =sin x222x; 3y + 2y = 1 ;e e x y(0) = 0 y (0) = 0y ; 2y00y5) y3)00007)y008)y00y000+y =ex x2)y00+ 2y 0 + y =0000000;+ 9y =9cos 3xy(0) = 1 y (0) = 0= 5 y= 4:10) y + 4y = 8 ctg 2x y449)0000y = ex(x ln jxj + C1x + C2)2) y = (C1 + C2 x)e x + xe x ln jxj3) y = (C1 + ln j sin xj) sin x + (C2 ; x) cos x4) y = sin 2x ln j cos xj ; x cos 2x + C1 sin 2x + C2 cos 2xcos 2x5) y = C1 cos x + C2 sin x ;cos x6) y = C1 cos x + C2 sin x + sin x ln j tg x=2j7) y = (1 + ln j sin xj) sin x ; x cos x228) y = ex ln+ e2x lnx1+e1+e x9) y = (1 + ln j cos 3xj) cos 3x + 3x sin 3x10) y = (5 + 2 ln j tg xj) sin 2x:1);;;||||{513; 23 ( J L54A 9y(n) + an;1y(n;1) + ::: + a1y0 + a0y = f (x):(23:1)- y :y = y + y = C1y1 + ::: + Cnyn + y(23:2) y1 ::: yn { ?2J L-4A, y { - (23.1)..
(23.1) f (x) f (x) = exNPm(x) cos x + Ql (x) sin x](23:3) Pm(x) Ql (x) { m l, y &, :{ + i , y y = exNP~k (x) cos x + Q~ k (x) sin x](23:4) P~k (x) Q~ k (x) { k = maxfl mg 9@{ + i r, y y = xr exNP~k (x) cos x + Q~ k (x) sin x]:(23:5)'9 P~k (x) Q~ k (x) y (23.1).+ ) y (. U22)@) y @514) (23.2):y = y + y :- (23.3) = 0 ..f (x) = exPm (x):(23:30)= (23.4) (23.5) y = exP~m (x)(23:40)( { ),y = xr exP~m (x)(23:50)( { r.) 23.1. 5 y000 ; y00 + y0 ; y = x2 + x:< ) Y :3 ; 2 + ; 1 = 02( ; 1) + ; 1 = 0( ; 1)(2 + 1) = 0:. 1 = 1 23 = i 1, 9yoo = C1ex + C2 cos x + C3 sin x:) f (x) = x2 + x f (x) = e0x(x2 + x):F = 0 , x2 + x 2, 9 y (23:40):y = e0x(Ax2 + Bx + C )515 A B C { 9. 50 = 2Ax + B y 00 = 2A y 000 = 0:y0 y 00 y 000 : y y ;2A + 2Ax + B ; Ax2 ; Bx ; C = x2 + x: 9 x x2;A = 1 9>>=x2A ; B = 1 > x0 B ; 2A ; C = 0 > A = ;1, B = ;3 C = ;1.
2, y = ;x2 ; 3x ; 1:) - y = yoo + y:y = C1ex + C2 cos x + C3 sin x ; x2 ; 3x ; 1: > 23.2. 5 y000 ; y00 = 12x + 6:< ) Y 3 ; 2 = 0 2( ; 1) = 0 1 = 0 2 2 = 1 1, 9yoo = C1 + C2x + C3ex:) f (x) = 12x + 6 f (x) = e0x(12x + 6):F = 0 , 12x + 6 { 1, 9 y (23:50) r = 2, = 0 m = 1 ..y = x2(Ax + B ) = Ax3 + Bx2 5160 = 3Ax2 + 2Bx y 00 = 6Ax + 2B y 000 = 6A:y00 y 000 : y6A ; 6Ax ; 2B = 12x + 6, 9 ,9x;6A = 12 = :x0 6A ; 2B = 6 J A = ;2, B = ;9. 2,y = ;2x3 ; 9x2:) - y = yoo + y:y = C1 + C2x + C3ex ; 2x3 ; 9x2: > 23.3.
5 y00 + y0 = 4x2ex:< ) Y 2 + = 0 ( + 1) = 0 1 = 0 2 = ;1, 9yoo = C1 + C2e;x:) f (x) = 4x2ex f (x) = e1x4x2:F = 1. = 1 , 4x2 { 2, y (23:40) = 1 m = 2 :y = ex(Ax2 + Bx + C ):51750 = ex (Ax2 + Bx + C ) + (2Ax + B )ex y00 = ex (Ax2 + Bx + C ) + 2(2Ax + B )ex + 2Aexy , ex 6= 0:2Ax2 + (6A + 2B )x + 2A + 3B + 2C = 4x2: 9 :8>>< 2A = 46A + 2B = 0>>: 2A + 3B + 2C = 0 A = 2, B = ;6 C = 7: 2,y = ex(2x2 ; 6x + 7):) - y = yoo + y:y = C1 + C2e;x + ex(2x2 ; 6x + 7): > 23.4.
5 y00 + 3y0 + 2y = x sin x:< ) Y 3 + 3 + 2 = 0 ( + 1)( + 2) = 0 1 = ;1 2 = ;2, 9yoo = C1e;x + C2e;2x:) f (x) = x sin x f (x) = e0x(0 cos x + x sin x):F + i = 0+1i = i , x { 1, 9 y (23.4) = 0 = 1 k = maxf0 1g = 1 ..y = (Ax + B ) cos x + (Cx + D) sin x5180 = (A + D + Cx) cos x + (C ; B ; Ax) sin xy00 = (2C ; B ; Ax) cos x ; (2A + D + Cx) sin x:y0 y 00 , y yN(A + 3C )x + 3A + B + 2C + 3D] cos x++N(;3A + C )x ; 2A ; 2B + 3C + D] sin x = x sin x:: 9 cos xsin x x cos x x sin x@ :9cos x3A + B + 2C + 3D = 0 >>>sin x ;2A ; 3B + 3C + D = 0 = :x cos xA + 3C = 0 >>>;3A + C = 1 x sin x3 , B = 17 C = 1 J 9 , A = ; 1050103D = 25 . 2,!!17133y = ; 10 x + 50 cos x + 10 x + 25 sin x:) - y = yoo + y:!!31713;x;2xy = C1e + C2e + ; 10 x + 50 cos x + 10 x + 25 sin x: > 23.5.
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