Методические указания к практическим занятиям по обыкновенным дифференциальным уравнениям (1012977), страница 3
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%,51, 5 :4z yn(1 n )1yn1.% 5 DG ..".#$%2. ( :z=yn .C 73.1yn1::= y1 n ,34. %. 26z = (1 n ) y n y47y =z yn(1 n )D: 4. 2. %7.5. ( 2;C6.C7. %: y1;::D8. 6 4.nz = ( x , C)2= ( x , C)42C132..2.7.% : y = y 4 cos( x ) + y tg ( x )$G$%)$:1.:%72 ;4;: y = y tg ( x ) + cos( x ) y 4:: f ( x , y) = a ( x ) y + b( x ) y n , 3: 7@2. ( :3.4. %y4 :C 7: z=:41=yy4 171 z y4=y3y4.y1= 3 tg ( x ) + cos( x )4yyz = 3 y3D3y =ytg( x ) + cos( x )4z = 3tg ( x ) z4: 4.2:z = 3tg ( x ) z 3 cos( x ) - G5.
( 2a ( x ) = tg ( x ) , b( x ) = cos( x ) , n = 4 ,:1-3 445:1z y43.".#$%. 27dz= 3tg ( x ) zdxdz= 3 tg ( x ) dxzln z = 3 ln cos(x ) + ln Cz = C cos 3 ( x )z = C( x ) cos 3 ( x )C ( x ) cos 3 ( x ) + C( x ) 3 cos 2 ( x ) ( sin( x )) = 3tg ( x ) C( x ) cos 3 ( x ) 3 cos(x )C ( x ) cos 3 ( x ) = 3 cos(x )C (x) = 31cos 2 ( x )C( x ) = 31€dx = 3tg( x ) + Ccos 2 ( x )z = ( 3tg( x ) + C)) cos 3 ( x ) -6.C7. y = 0 - 48.
"2: y 3 = ( 3tg( x ) + C) cos 3 ( x ) -;:2.: y 3 = ( 3tg ( x ) + C) cos 3 ( x ),y=02@.".#$%. 282.2.7. )..$!$%H) >F%&$ ! #%$%)1 # = >%&( )..$!$%H) > (M ( x , y)dx + N( x , y)dy = 0 ,=!$ $>$%)$ 2.9.,5374:C445F( x , y) .LC4,N ( x , y).xM ( x , y)dx + N( x , y)dy = 00$* %)$.3M ( x , y)y:24F( x , y) = C5,5F( x , y)= M ( x , y)x:F( x , y)= N ( x , y) .y41.
%N ( x , y),x2. 6 45453F( x , y)= M ( x , y)xC)F( x , y)= N ( x , y) .y452; F( x , y) , 445 :4: F( x , y) = C2)C'M ( x , y)dx + N( x , y)dy = 05M ( x , y)y3.55;C);.;1:344:C).:C; 734yx:) F( x , y) = M ( x , y)dx + C( y)C) F( x , y) = N( x , y)dy + C( x )4. %4%775.C) 4:)4477D;4::; 5 M ( x , y)7) 4; 5 N ( x , y):y,::7:C (x) .C) 4x.C ( y) ,.".#$%. 29) C( y) = C ( y)dyC) C( x ) = C ( x )dx6. 6 45; F( x , y) .7. 6 4C12.2.8.% : 2x y dx = ( y 2x 2 )dy$G$%)$:M ( x , y)dx + N( x , y)dy = 0 : 2 x y dx + ( y 2 + x 2 )dy = 0123142431. 6 4 2M ( x ,y):%7M ( x , y)= 2xyM ( x , y)y2. @53.F( x , y) :N ( x ,y)N ( x , y)= 2x .xN ( x , y),: 7x4F( x , y) = C , 32.F( x , y)= M ( x , y)xF( x , y)= N ( x , y) .yF( x , y)= M ( x , y) :xF( x, y) = 2x y dx = 2 y x dx = 2 yx2+ C( y )2F( x, y) = y x 2 + C( y)4.
%N ( x , y) , 4477D4y:F( x, y)= x 2 + C ( y)y:x 2 + C ( y) = y 2 + x 2C ( y) = y 25.C( y ) =y 2 dy =y3(/334 2!)4.".6. 6 4 25: y x7.#$%;y3=C 32y33;: F( x , y) = y x 2524.2.3. )..$!$%H) >F%&$ ! #%$%)1 1"% )"$>F% =! )L#1-3 4D5354C :474D:42,= !1 + , %$! L!$G$%%&$% ': 2C. 30::5,7:3y .54; 5D5:47 5::.:: y = f i ( x, y), i = 1,2..5.% : ( y ) 2 + xy = y 2 + xy$G$%)$:( y ) 2 + xy = y 2 + xy(y ) 24xy = y 25xy411({y ) 2 2{yx + x224{a {a2b(yy%7,%3y = xe xex + C"1 2x) = ( y21x=y2:1 211x ={y 2 2 {y x + x 2424{a {a2by: 21x = (y24-2) y = y-:y = y4y = xe x1 2x4b21 2x)21x21) y = y: y = Ce x ,b27ex + C7;1:1x)2:4y = Ce x , 43y = y47.".#$%C147:73,7 5142,57: ;,5 34:$,: 235D(y : y = f (x, y ),1.
( : 2$1. ( : 2x:x = f ( y, y )2.p=y =4dy,dx34y = f ( x , p)44:7L:1-3 4dx =5 ,47,: 4x = ( p, C )6.234x = (p, C)y = f ( (p, C), p)4D48.: 4. 2:5 ,4:32: 2:7 5:x = f ( (p, C), p)y = (p, C)237y = ( p, C )6 47. %f ( y, p)f ( y, p)dx +dpyp1-3 4:4::dy,4pL( 27dyf ( y, p)f ( y, p)=dx +dppyp: 25.,47f ( x , p)f ( x , p)dx +dpxppdx =:7dx =4.dy = pdx , 47x = f ( y, p )f ( x , p)f ( x , p)dx +dpxpdy =x : x = f ( y, y ),y = f (x, y ):G)8y:3.y.x.,8.
31:44;2x = .... 6: y = ....56 4C1p = .... , 42;45.".#$%. 322.9.% : y = ln(1 + ( y ) 2 )$G$%)$:1.: 2y ,: 2y:y = ln(1 + ( y ) 2 )2.p=y =43.:dy,4dx: y = ln(1 + p 2 )744dy = pdx , 4:pdx =7:12p dp1+ p22dp - 41+ p2dx =5. ( 2:12p dp1+ p2dy =4.77:;14:dx =2dp1+ p2x = 2 arctg(p) + C -6. 6 4 2232:4;17 54:x = 2 arctg(p) + Cy = ln(1 + p 2 )7. %:D448. "7D:y = ln(1 + 0 2 ) = ln1 = 0: 0 = ln(1 + (0 ) 2 )y=0%: p=02x = 2arctg(p) + C,y = ln(1 + p 2 )740=0( D), : 7y=0 -y=02.10.% : y ( x ln y ) = 1$G$%)$:1.: 2x=1+ ln yyy ,: 2x:2..".2.3.#$%p=y =4:7: x=41+ ln pp4dx = (4.dy,4dx7:1 1+ )dpp2 pdx =:. 33dy,4p7:1 1dy= ( 2 + )dppppdy = (5. ( 21+ 1)dp - 4p:;14:;14:dy = (1+ 1)dppy = ln p + p + C 6. 6 4 2723247 5:1+ ln ppy = ln p + p + Cx=7.%:D,48.
"1+ ln pp:y = ln p + p + Cx=422.:p=0x=l+ ln 0 0:D,.".#$%3.. 346O P6n -3 45C17:F( x, y, y , y ...y ( n ) ) = 0y ( n ) = f ( x, y, y , y ...y ( n1))-,: 2( x, y, C1 , C 2 ...C n ) = 0( 23.1.> #)14:4::.4:E$ "# # %)1 ) $ )% "#$%% ") !$G$%)1)..$!$%H) >F%&( ! #%$%)' #& G)( = !1 + #=!$ $>$%)$ 3.1. 6,7;1 3 n/ 2:7,37=!$ $>$%)$ 3.2.n 1455 ;7%:7 / 2:n -4– G,:. 6 47DG4-3 4: 75:C55.5; y75 x 0 ,34:2:7.P0 = ( x 0 , a , b, c,..., s) ,57C17..%:7 / 2 .y IV = f ( x, y, y , y , y )y( x 0 ) = ay (x 0 ) = by (x 0 ) = cy (x 0 ) = d$ !$0G). ( $ !$0: y ( n ) = f ( x, y, y , y ...y ( n5f ( x, y, y , y ...y ( n;42E$ "# # %)1 ) $ )% "#$%% ") !$G$%)11))C1))D,D 144)35,77533 ,7.".#$%x0hx 0 + h, h > 0 ,xy ( n ) = f ( x, y, y , y ...y ( n1)),.
351;127:y( x 0 ) = ay (x 0 ) = by (x 0 ) = c.......y (n1)(x 0 ) = s420$* %)$:137D7:4:7f fff,,... ( ny y yyy:CC1)545 C3 43 C5 43,D."#$!J $%)$:1. ( 21,2. ( 27D,C4f fff,,... ( ny y yy:3.2. > * ) )%"$ !)! # %)$D; 1C1 31;G74.74#& G)( = !1 + #2 3 45:D2:n533.1.% : y = sin x$G$%)$:D)1).'31)5557: y(n) = f (x) .) 1.72f ( x, y, y , y ...y ( n53y dx = sin xdx;4;7:y = cos x + C1y dx = ( cos x + C1 )dxy = sin x + C1x + C 234..".#$%y dx = ( sin x + C1x + C 2 )dxy = cos x + C1. 36x2+ C 2 x + C32x2+ C 2 x + C3"#$": y = cos x + C127:."<)1. %242.
( 25C5:44:C ;4:1 $C:,$5 (4.1-2).3..C4C:4:45 .2. BDD.5. %46. 6 445 .1-3 4n4; 11-3 ,1-3 43.4.4$ ,:22 32..) 2.: F( x , y ( k ) , y ( k +1) ,..., y ( n ) ) = 0 -Dy.:5: z( x ) = y ( k ) ,3y ( k +1) = z. .n 4:D.".#$%. 373.2.% : ( x 2 + 1) y = 2 x y$G$%)$:1.2-3 44%:7: y = z( x ) ,:: ( x 2 + 1)z = 2 x z -2. ( 24z =75 ,D 1y =z .31-3 41-3 42xz -Gx 2 +1y.5 .5 ::;14.dz2x= 2zdx x + 1dz2x= 2dxzx +1z = C1 ( x 2 + 1) 3.C22d ( x 2 + 1)x 2 +11-3 45: y = C1 ( x 2 + 1) -;:(ln z =47341-3 4:y=Cy = C1 x 2 + C1y=Cy = (C1 x 2 + C1 )dx4; 1272: y = C1) 3.x3+ C1 x + C 2 .3.x3+ C1 x + C 23x3+ C1 x + C 23.y = C127: y = C15. %6.
"7720y =0:4C1 :C1y = C1:5: 7C1 = 04. %ln z = ln( x 2 + 1) + ln C12x3+ C1 x + C 23y = C1y =C,x3+ C1 x + C 2 , 4 G34C1 = 04 225.".#$%: F( y, y , y ,...., y ( n ) ) = 0 -. 38Dx.:4: y: p ( y) = y ,5y = (y ) =3dy dp( y) dp dy===p y =p pdxdxdy dx3.3.% : (y )2 + 2y y = 0$G$%)$:1.2-3 44%:7: y = p( y ) ,:: p 2 + 2y p = 0 -2. ( 24p =75 ,x.y = p p.31-3 45 .1-3 41p -G2yD 15 ::;14.dp1=pdy 2 ydp1 dy=p2 yp=3.C1yC2;:ln p =1-3 4: y =C1y1ln y + ln C1251-3 45ln p = ln1+ ln C1y.".#$%(24734:C1 :: 7C1 = 0C1y =00C1yy =y=C.
39dy C1=dxyydy = C1dx32 2y = C1 x + C 2334.:477275. %2: y 2 = C1 x + C 23y=C:3 3 .2.36. "; 1:2 2y = C1 x + C 2 ,3y=C) 4.1,y: 7 ,:k y, y. ., 3k y:k – ;C7,0.:5: z( x ) =yyy =z y,3y = ( z y) = z y + z y = z y + z ( z y) = z y + z 2 y3.4.% : x y yx (y )2 = y y$G$%)$:1.2-3 45 ,Dxy.Gy.".#$%. 40: x (ky) (ky ) x (ky ) 2 = ky ky - 5 D%k2 ,D4%:755(k0 ), 43y = z y + z2 y: y = z( x ) y ,:73.:x y (z y + z 2 y) x (z y) 2 = y z yx z y2 + x z2 y2x z2 y2 = y2 zx z + x z2x z2 = zx z =z -1-3 42.
( 24z =751-3 41z -Gx:5 :;14.dz 1= zdx xdzdx=zx3.C(z = C1 x -ln z = ln x + ln C1;:24:7y= C1 xy3y = C1 x y 4:: 7C1 = 0y = C1 x yy=Cdy= C1 x ydxln y = C14.:475. %6. ":1-3 47273 3 .2.ln y = C1x2+ C2 ,2y=C1-3 455C1 :C1y =020dy= C1 xdxyx2+ C22: ln y = C1x2+ C22y=C:.".#$%3.3. $G$%)$ L1. ( 2*).6 4G) >1#& G)( = !1 + #=n-32. %C1C13. %C13 C22$.:.n247 54. ( 2,..
41:47:. %C1 , C 2 ...C n .4C1 , C 2 ...C n .5. %: 7C12–G2:7 / 2 .3.5.% : ( x 2 + 1) y = 2 x yy(1) = 1y (1) = 2$G$%)$:1. ( 2C1(.43.2.).: y = C122.C1x3+ C1 x + C 232::y = C1 x 2 + C13. %:7y(1) = 1 : 7y (1) = 2 : 7%74. ( 2,7,711 = C1 + C1 + C 2:32 = C1 + C1:C1x =144x =12y =1, : 755y =2, : 7741 = C1:;:13+ C1 1 + C 232 = C1 12 + C1.".#$%11 = C1 + C1 + C 232 = C1 + C15. %/ 2": y=x31+x33C1 = 141 = C1 + C 232 = 2C1: 7C1.
42C2 =213: y =1x31+1 x332:7.".#$%. 434.6: p n (x) y (n ) + p n 1 (x ) y (n=!$ $>$%)$ 4.1.:1)+ ... + p 2 ( x ) y + p 1 ( x ) y + p 0 ( x ) y = f ( x ).6p n ( x )...p 0 ( x )545=!$ $>$%)$ 4.2.50,f (x )D f (x ) = 0 ,D2 ...D 5 (a , b ) .:,.=!$ $>$%)$ 4.3. +1,4:1 y1y1 ( x ), y 2 ( x )...y n ( x )5+2y2+ ... +n yn:=0;:45(a , b ) ,7 , 5 3=0n5 4=!$ $>$%)$ 4.4.3:n4D 5:(a , b )22(+ () G 3.$ !$0"! +" !$ 2E$!$G$%)1 >)%$'%y1 ( x ), y 2 ( x )...y n ( x ) - n3,:C1C1 ....C n – 4$ !$0:!$G$%)1 %$3;1D 53% !3;1 3! #%$%)1(a , b ) 724:.2,24"! +" !$ 2E$C14%( x ) = C1 y1 ( x ) + C 2 y 2 ( x ) + ... + C n y n ( x ) ,y3% !%! #%$%)14y.C1 3(x )7223.".#$%4.1.
)%$'%&$% !%&$. 44= " 1%%&0) + Q..)H)$%" 0) (=!$ $>$%)$ 4.5.45 G)::a n y (n) + a n 1y (n31)+ .... + a 2 y + a 1 y + a 0 y = 0– a n , a n 1 .....a 1 , a 07+"1.57 5an2.n53.+ (5 D+ an551( 3 Cn 1+ .... + a 27 5 33: 4y3n(B )27 5 ):+ a1 + a 0 = 0:21,2 ... n .:( x ) = C1 y1 ( x ) + C 2 y 2 ( x ) + ... + C n y n ( x ) ,5y j (x);j.>;)j–45./' 3C)2j4: 4 21,: C je52–5: 4 2,k53–5./' 3CB .(5:.jxk).D3757C1: (k: (C j + C j+1 x + C j+ 2 x 2 + ... + C j+ k 1 x k 1 ) ejx:)..".j)=j+1:#$%+i=–4i4–7./j=+i' 3=j+1=–4i./j=72,> 0-:5235- 4 D754 5: 4 2:43x5=x(55k),D ki5:.+ C j+1 sin( x ) eCj,D 1i- 4 D52k=5,.: C j cos( x ) e- 4 Dk5Cj54 5+i' 31: 4 2+i4 5- 4 D72j3)4 55. 455:.:(C j + C j+1 x + C j+ 2 x 2 + ...
+ C j+ k 1 x k 1 ) cos( x )ex+(C m + C m +1 x + C m + 2 x 2 + ... + C m + k 1 x k 1 ) sin( x )e4.1.% : y +y2y = 0$G$%)$:1.5L7 552. ( 2,57 5D = b21, 23. 6 4 21=:2+2=0ax 2 + bx + c = 0 , 3:4 a c = 1 4 1 ( 2) = 9b± D1± 9=2a22= 2 -4:51= 22=1a = 1, b = 1c = 2.x.".2%#$%=1 - 452x: y = C1 e7+ C 2 e1 x= 21: y = C1e". 462x2=1+ C2e x4.2.% : 4y + 4y + y = 0$G$%)$:1.52. ( 27 55: 47 52+ 4 +1 = 0:D = 16 4 4 1 = 0=1, 23.
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