1. Интегралы ФНП Диф_ур (853736), страница 34
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. . 24.7. J 8>dx>< dt = x ; 5y(24:17)>dy = 2x ; y:>:dt< ) : !1;5A = 2 ;1 ;5 = 0 1 ; 2;1 ; (1 ; )(;1 ; ) + 10 = 02 + 9 = 0:'p ( )p12 = ;9 = 3i { -.) 5 != f1 2g A, = 3i (A ; 3iE ) ! =!o :1 ; 3i ;5 ! 1 ! = 0 ! ( (1 ; 3i)1 ; 52 = 02 ;1 ; 3i 2021 ; (1 + 3i)2 = 0:: 9 ( 9 0). 1 = 5! 2 = 1 ; 3i = f5@ 1 ; 3ig: 5 (24.17) !! 3it53it :e =e(24:18)1 ; 3i535123- (24:18) , C eit = cos t + i sin t :5 ! e3it =5e3it ! =5 cos 3t + 5i sin3t ! =1 ; 3i(1 ; 3i)e3it(1 ; 3i)(cos 3t + i sin3t)! h5 cos 3t + 5i sin 3t2 = ;1 i ==icos 3t ; 3i cos3t + i sin 3t ; 3i2 sin 3t)!5 cos 3t + 5i sin3tcos 3t + 3 sin 3t + i(sin 3t ; 3 cos 3t) =5 cos 3t ! + i5 sin 3t ! :cos 3t + 3 sin 3tsin 3t ; 3 cos 3t) 5 cos 3t !cos 3t + 3 sin 3t 5 sin 3t !sin 3t ; 3 cos 3t (24.17) , , (24.17):x! = C5 cos 3t ! + C5 sin 3t ! 1 cos 3t + 3 sin 3t2 sin 3t ; 3 cos 3tyx = 5C1 cos 3t + 5C2 sin 3ty = C1(cos 3t + 3 sin 3t) + C2(sin 3t ; 3 cos3t): >+.
+ . 24.8. J 8 dx>>< dt = 3x + 2y(24:19)>dy>:dt = ;2x ; y:< ) S !32A = ;2 ;1 536 2 = 0 3 ; ;2 ;1 ; (3 ; )(;1 ; )+4 = 0 2 ; 2 +1 = 0 ( ; 1)2 = 0:= = 1 { .) V (24.19) 8< x = (1 + 1t)e1t(24:20): y = (2 + 2t)e1t:5dx = et + ( + t)et dy = et + ( + t)et1122dt 1dt 2 (24.19):8< (1 + 1 + 1t)et = N3(1 + 1t) + 2(2 + 2t)]et(24:21): (2 + 2 + 2t)et = (;2(1 + 1t) ; 2 + 2t)et:2 et 6= 0, 9 t :9=t 1 = 31 + 22t 1 + 1 = 31 + 22 9=t 2 = ;21 ; 2t 2 + 2 = ;21 ; 2 8< 1 = ;2: 1 = 21 + 22:1. 1 = 2 = 0, 1 = ;2 , , ,1 = 1 2 = ;1: 1 = 1 2 = ;1 1 = 2 = 0 (24.21), (24.19)8< x = et: y = ;et :5372. 1 = 1, 2 = ;1 21 + 22 = 1: : , , 1 = 0, 2 = 12 .
1 = 0, 2 = 12 , 1 = 1, 2 = ;1 (24.21), (24.19)8< x = tet: y = 12 ; t et:) F 8< x = C1et + C2tet: y = ;C1et + C2 21 ; t et:> &,$ .1)8 dx>>< dt = 8y ; x>>: dy = x + ydt2)8 dx>>< dt = 4x ; 5y>>: dy = x x(0) = 0 y(0) = 1dtx = 2C1e3t ; 4C2e 3t y = C1e3t + C2e 3tx = ;5e2t sin t y = e2t(cos t ; 2 sin t)3) x = (C1 + C2 )e3t y = (C1 + C2 + C2t)e3t:1)2);;||||{5383)8 dx>>< dt = 2x + y>>: dy = 4y ; x:dt "" 11. 5 :Zdx :px x2 + 12. 5 :Z(4 ; 3x)e;3xdx:3. 5 :Z x3 + 6x2 + 13x + 9(x + 1)(x + 2)3 dx:4. : , :y = (x ; 2)3y = 4x ; 8:5. : , :(x = 5(t ; sin t)y = 5(1 ; cos t)0 t :539 21. 5 :Z 1 + ln xx dx:2.
5 :Zparctg 4x ; 1dx:3. 5 :Z x3 + 6x2 + 13x + 8dx:x(x + 2)34. : , :py = x 9 ; x2y = 0 (0 x 3):5. : , :(x = 3(2 cos t ; cos 2t)y = 3(2 sin t ; sin 2t)0 t 2:540 31. 5 :Zdx :px x2 ; 12. 5 :Z(3x + 4)e3xdx:3. 5 :Z x3 ; 6x2 + 13x ; 6(x + 2)(x ; 2)3 dx:4. : , :y = 4 ; x2y = x2 ; 2x:5. : , :(x = 4(cos t + t sin t)y = 4(sin t ; t cos t)0 t 2:541 41. 5 :Z x2 + ln x2x dx:2. 5 :Z(4x ; 2) cos 2xdx:3. 5 :Z x3 + 6x2 + 14x + 10(x + 1)(x + 2)3 dx:4.
: , :y = sin x cos2 x y = 0 (0 x 2 ):5. : , :(x = (t2 ; 2) sin t + 2t cos ty = (2 ; t2) cos t + 2t sin t0 t :542 51. 5 :Zpx4 +xx2 + 1 dx:2. 5 :Z(4 ; 16x) sin 4xdx:3. 5 :Z x3 ; 6x2 + 11x ; 10(x + 2)(x ; 2)3 dx:4. : , :py = 4 ; x2y = 0 x = 0 x = 1:5. : , :(x = 10 cos3 ty = 10 sin3 t0 t 2 :543 61. 5 :Z (arccos x)3 ; 1p1 ; x2 dx:2. 5 :Z(5x ; 2)e3xdx:3. 5 :Z x3 + 6x2 + 11x + 7(x + 1)(x + 2)3 dx:4. : , :py = x2 4 ; x2y = 0 (0 x 2):5. : , :(x = et(cos t + sin t)y = et(cos t ; sin t)0 t :544 71.
5 :Ztg x ln cos xdx:2. 5 :Z(1 ; 6x)e2xdx:3. 5 :Z 2x3 + 6x2 + 7x + 1(x ; 1)(x + 1)3 dx:4. : , :y = cos x sin2 x y = 0 (0 x 2 ):5. : , :(x = 3(t ; sin t)y = 3(1 ; cos t) t 2:545 81. 5 :Z tg(x + 1)cos2(x + 1) dx:2. 5 :Zln(x2 + 4)dx:3. 5 :Z x3 + 6x2 + 10x + 10(x ; 1)(x + 2)3 dx:4. : , :py = ex ; 1y = 0 x = ln 2:5. : , :81 cos t ; 1 cos 2t>>x=><24>11>>: y = sin t ; sin 2t24 t 2 :23546 91. 5 :Zx3 dx:(x2 + 1)22. 5 :Zln(4x2 + 1)dx:3. 5 :Z 2x3 + 6x2 + 7x + 2dx:x(x + 1)34. : , :y = p 1 y = 0 x = 1 x = e3:x 1 + ln x5.
: , :(x = 3(cos t + t sin t)y = 3(sin t ; t cos t)0 t 3 :547 101. 5 :Z 1 ; cos x(x ; sin x)2 dx:2. 5 :Z(2 ; 4x) sin 2xdx:3. 5 :Z x3 ; 6x2 + 13x ; 8dx:x(x ; 2)34. : , :y = arccos xy = 0 x = 0:5. : , :(x = (t2 ; 2) sin t + 2t cos ty = (2 ; t2) cos t + 2t sin t0 t 3 :548 " " 11. 5 :z = x arctg xy :2. 5 A:4x3 + y3 ; z 2 + 3xyz = 0A(1 0 2):3. 7 9 :z = 2y3 + x2 + 6xy + 18y2 + 18x + 54y + 54: 21. 5 :vu33u3 x + yxtz=:y2. 5 A:z = x4 ; 4y4 + 3yx2 ; x + 1A(2 1 23):3. 7 9 :z = ;6y3 + 54x2 ; 6xy ; y2 ; 150x ; 14y:549 31.
5 :z = yex=y :2. 5 A:z = x2 ln(x + 4y2 ) + x3y2 ; y + 3A(2 0 3 + ln 16):3. 7 9 :z = 3y3 ; 2x2 ; 12xy + 27y2 ; 36x + 81y ; 15: 41. 5 :z = x ln cos(xpy):2. 5 A:z = sin(x2 + y + z ) ; x2yz ; 1A(0 1 ;1):3. 7 9 :z = ;2y3 ; x2 ; 6xy ; 18y2 ; 12x ; 36y + 9:550 51. 5 :y:z = x1arctg+ x22. 5 A:xy + yz ; 3xyz = 2A(1 2 0):3. 7 9 :z = ;3x3 + 81x2 ; 12xy ; 2y2 ; 93x + 32y + 4: 61. 5 :01xz = 2x @arctg py A :2. 5 A:3z 2 = 4ex+y ; 3xy2z 3 + 2A(1 ;1 1):3. 7 9 :z = 6y3 + 18x2 + 6xy + y2 ; 6x + 10y + 13:551 71. 5 :!x2z = y cos y :2.
5 A:z = 4 tg2 xy ; 3y3 + x3 ; 1A(2 0 7):3. 7 9 :z = ;y3 ; 3x2 ; 6xy ; 3y2 + 3y + 2: 81. 5 :z = y arccos(xpy):2. 5 A:x2yz 3 + 4y2 = ez + 15A(1 2 0):3. 7 9 :z = ;2x3 + 6x2 + 6xy + y2 + 12x + 24:552 91. 5 :vu3+xuxytz=:y22.
5 A:4z 2 = x2y3 + cos(x + y2) + 14A(;1 1 2):3. 7 9 :z = ;2x3 + 24x2 + 6xy + 3y2 ; 90x ; 18y + 1: 101. 5 :pz = ln( x 2cos(x y) ):22. 5 A:yz 3 ; x2 z; y = 2z 2 A(2 3 1):3. 7 9 :z = ;2x3 + 6x2 ; 6xy ; y2 + 8y ; 1:553 " " 11. 5 :2y0 = xy 2 + 4 xy + 2:2. 5 ' :y0 ; xy = x2 y(1) = 0:3. 5 :y000 x ln x = y00 :4. 5 :y00 ; 2y0 = 2 ch 2x: 21. 5 :32xy0 = 3y2y2++2yxx2 :2. 5 ' :y0 ; y ctg x = 2x sin xy(=2) = 0:3. 5 :xy000 + y00 = 1:4. 5 :y00 + y = 2 sin x ; 6 cos x + 2ex:554 31.
5 :+ y:y0 = xx ;y2. 5 ' :y0 + y cos x = 21 sin 2x y(0) = 0:3. 5 :2xy000 = y00 :4. 5 :y000 ; y0 = 2ex + cos x: 41. 5 :qxy0 = x2 + y2 + y:2. 5 ' :y0 + y tg x = cos2 xy(=4) = 1=2:3. 5 :xy000 + y00 = x + 1:4. 5 :y00 ; 3y0 = 2 ch 3x:555 51. 5 :22y0 = xy 2 + 6 xy + 3:2. 5 ' :y0 ; x +y 2 = x2 + 2x y(;1) = 3=2:3.
5 :tg x y00 ; y0 + sin1 x = 0:4. 5 :y00 + 4y = ;8 sin 2x + 32 cos 2x + 4e2x: 61. 5 :3 + 4yx23y0xy = 2y2 + 2x2 :2. 5 ' :y0 ; x +1 1 y = ex(x + 1) y(0) = 1:3. 5 :x2y00 + xy0 = 1:4. 5 :y000 ; y0 = 10 sin x + 6 cos x + 4ex:556 71. 5 :y0 = x2x+;2yy :2. 5 ' :y(=2) = 1:y0 ; xy = x sin x3. 5 :y000 ctg 2x + 2y00 = 0:4. 5 :y00 ; 4y0 = 16 ch 4x: 81. 5 :xy0q= 2 x2 + y2 + y:2. 5 ' :y0 + xy = sin x y() = 1=:3. 5 :x3y000 + x2y00 = 1:4.
5 :y00 + 9y = ;18 sin 3x ; 18e3x:557 91. 5 :23y0 = xy 2 + 8 xy + 4:2. 5 ' :y0 + 2yx = x2 y(1) = 1:3. 5 :tg x y000 = 2y00 :4. 5 :y000 ; 4y0 = 24e2x ; 4 cos 2x + 8 sin 2x: 101. 5 :32xy0 = 32yy2++63yxx2 :2. 5 ' :22x2x0y + 1 + x2 y = 1 + x2 y(0) = 2=3:3. 5 :y000 cth 2x = 2y00 :4. 5 :y00 ; 5y0 = 50 ch 5x:558 !"#$%"&'(%%") *(+*"##"1. . 5 , .2. F .3.
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