Методические указания к практическим занятиям по обыкновенным дифференциальным уравнениям (1012977), страница 2
Текст из файла (страница 2)
C15 ;7::C,42.74,5 54;:3::, 4:;C4;:52.,47C1 , C 2 ...C n . 6 75:: C1 34:..2-3 4C12K5 : y +y=0: y = C1 cos x + C 2 sin x323, 474y ( 0) = 17y ( 0) = 2 :y = cos x + 2 sin x=!$ $>$%)$ 1.853D 5C24:3 ,543;12,5 D2.75.".#$%2..86 1-C11-3 465 :F( x , y, y ) = 0(1)y = f ( x , y)(2)K1-3545:D 1:dy :M ( x , y)dx + N( x , y)dy = 07,7(2: 435 4dy= f ( x , y)dxy = f ( x , y)5dy = f ( x , y)dx3)3CM ( x , y) + N ( x , y)G :f ( x , y)dx + ({1) dy = 0123N ( x,y)M( x,y)M ( x , y)dx + N( x , y)dy = 0,4 5 Ddy=0dxdy=dxM ( x , y)N ( x , y)y =M ( x , y)N ( x , y)14243f ( x,y)C121-3 4y = ( x , C)55D);:3:,475,732y = ( x , C) . ' 3 ,;C75 P 45 G(533f ( x, y) .5453, : 75 D45775 45=!$ $>$%)$ 2.1. !4; 45 y = f ( x , y)1-3 44545:= !1 +7 5 3( x, y) 541-C:354( x , y, C ) = 0 .$ 0$"!)*$ +)' 0& >%D3( x, y) , 3347 5:.4554.
%53f ( x, y) .7 5: 5C :45( x, y) ,54.".2.1.#$%> #)1E$ "# # %)$ ) $ )% "#$%% "F !$G$%)1)..$!$%H) >F%%2! #%$%)1 1-4 5 7 52,5:;1:= !1 + .*7, 5 5 4,44:5;7:.9,5G)CC1 ,.57C 7;5G; y( x )75 x 0 .=!$ $>$%)$ 2.2. 627/ 2,% 51-3;1 35 :5 ;72:7C; y( x 0 ) = y 0 , 377 / 21:,4;155:57x 0 , y0 - :.,:7534;:C.$ !$0G).
( $ !$0E$ "# # %)1 ) $ )% "#$%% ") !$G$%)1)% : y = f ( x , y) ,54f ( x, y)5Dx0hxx 0 + h, h > 0C;42D 1,75P0 = ( x 0 , y 0 ) ,71y = f ( x , y) ,2; y( x 0 ) = y 0 .;1124GD4C0$* %)$.421D,37D4C;,7.:4,545C3 Cfy:CD."#$!J $%)$.1)52f ( x, y).4,x0hxx 0 + h, h > 01.".2)#$%5G 3 ,1C3,7G. 104742fy:..%:1) y =4:7 / 2y, y( 0) = 1x2) y = x y , y(1) = 0.1) (7 y =:(44y, y( 0) = 1 .x: f ( x, y) =;7G55: P0 = (0,1) .7,72) (:(42.5P0 = (1,0) .4; 44, 74, ,4, 752.:;75 ,75.:5fx=y 2 y5GC4:,,.5352- G5.5D7:.7 / 257775 P0 = (1,0),7::::1 = 0757 / 27457: f ( x, y) = x y - G,74; 7:4: y=C x,3G7(P0 = (0,1)77 y = x y , y(1) = 0 .4:::.2C175..2C1475 ,7547 / 257, 7,::y-Gx, ,575,P0 = (1,0) ..".#$%,45y=x24C1237x2y=+ C, y = 0 ,4:C1. 1124772, 747/ 2 ::1, y=042.2 )..$!$%H) >F%&$ ! #%$%)1 1"% )"$>F% =! )L#= !1 + , ! L!$G$%%&$% '2.2.1 )..$!$%H) >F%&$ ! #%$%)1 ! L $>1ME)0) 1 =$!$0$%%&0)y = f ( x , y)=!$ $>$%)$ 2.3.:;14,3:D45:P( x ) Q( y)dx + R ( x ) S( y)dy = 0y = (x )%7( y)4:5y,431.
%552. %D,:5x., 5 ::5 :54.dy.dxy3.:5C5dx . %75dy ,y:447D–, 7 C55D72; – 4x. /.4. %3; 735. %C : 4:D4544y,.( :2.(43.67–)2.x.".#$%6. 6 4C1.422.23C35:,44. 12C42.' 5C 1 ;1D74,5 15 : 24:.2.1.% : x y + ( x + 1) y = 0$G$%)$:1.:4:2 ;47x,3;: y =x yx +1D,D3xx +1{{y - G:3. ( :4:::34 y, 4x +1dxx +11dxx +1ln y = x + ln x + 1 + C -4;14dyx=dxy x +1;7ln y =C :;dyx=ydx x + 1dyx=dxyx +1%5( y)(x)2.4y:y =4.
%:72; 4 x:ln y =x +1 1dxx +1ln y =1 dx +1d ( x + 1)x +12:ln y = x + ln x + 1 + ln Cln y=ey=exex + ln x +1 + ln C( x + 1) C -y=e2xeln x +1e ln C3.".#$%5. %::51D424;15) x +1 = 047;; x = 1, 4,4:y=074(Dx= 1 -), : 7.; y =0, 45x 0 + ( x + 1) 0 = 07::0+0 = 07420=0( Dxy=0 -), : 75 57: y=e6. "D:5D4 2, 4y+0=0C) y = 0 . %%2x = 1. %2G 351 y + ( 1 + 1) y = 04.. 1372,C=04( x + 1) C2.2.y 2 + 1 dx = x y dy% :$G$%)$:1.4:y =dy,dx::2 ;4:7D4x,331x{(x):;: y =:D:y2 + 1-Gy14243( y)dy 1=dx xy2 +1y:;:y2 +1 = x y y .y2 + 1x y,4y:y =2.2 ;4;145;3G.".#$%3. ( :44. %:3y dyy2 +1=1dxx;7y dyy2 + 1=4 y, 4::51D545y 2 + 1 d0 = 0 y dy47,.25G 3= ln x24 2, 4D;,4:; x = 0, 4:0=07424( D4), : 75 5x=0 -: 74y=0 -22:.C) y = 0 . %50 2 + 1 dx = x 0 d0%7:7; y =0, 424.::dx = 0y2 +1 = 0)6.
"2DGy2 +1y 2 + 1 = ln x + C -4) x = 0. %d ( y 2 + 1)124;1; 4 x:1dxx12 y 2 + 1 = ln x + C25. %. 14y 2 + 1 = ln x + C, x = 00=0(4D5 5), : 7: 7y, :..".#$%2.2.2. )..$!$%H) >F%&$ ! #%$%)1, =!)#. 151E)$ 1 + ! #%$%)10! L $>1ME)0) 1=$!$0$%%&0)y = f (ax + by + c):;1% 5 D441 ;:z (x) = a + b y (x)y =DC45: z( x ) = ax + b y( x ) + c .G :zab312=zabf{(z)f ( ax + by + c )yz = a + b f (z) -:;14dz= a + b f (z)dxdz= dxa + b f (z).7: 4C2.2.3.% !44C3;45n,M ( x , y):, 4).= !1 +:k>0;C 3 745 (:%&$ )..$!$%H) >F%&$ ! #%$%)1 1-=!$ $>$%)$ 2.4. +4, 5 353: M (kx , ky) = k n M ( x , y):M ( x , y) = x 2 + xyM (kx , ky) = (kx ) 2 + kx ky = k 2 ( x 2 + xy) = k 2 M ( x , y) M ( x , y)dx + N( x , y)dy = 0 ,=!$ $>$%)$ 2.5.45 ,355M ( x , y)D4.N ( x , y);42.:1-35.".#$%1-3 45f ( x , y) –DC51-3 4.y =5D35DC4.
16: y = f ( x , y) , 35 50-4.yx:y =f4x+yx y123f ( x ,y)6f ( x , y) -l553kx + ky x + yf (kx, ky) ==kx ky x y:71+ yxy =y1x123f;1544yy=z xxz1x4+3z = f{(z)42yf1xdz1= (f ( z ) z)dxxdzdx=(f ( z ) z ) x:yxy =z x+z:;11-5z( x ) =yxz = (f ( z ) z ):4G .z( x ) =2.::1 ; :.% 5 D1. %47x:yx1-3 4:0-, 5 :y= z( x ) , y = z x + zx4$4.DC45.".#$%3. %4:7;14. ( 2445.Dz = (f ( z ) z )71- Gx.7.C6.
%, 4. 17;::.D7. 6 442C12..2.3.% :y ln y ln x=yx$G$%)$:1.::2 ;4%747y =;: y =:3y(ln y ln x )x4Cyyln - Gx 23x11-3 455yf( )x2.3. %: z=:7:y =z x+zz x + z = z ln zz =4. ( 2y,x1(z ln z z) x:;1:4dz 1= (z ln z z)dx xdzdx=z(ln z 1)xd (ln z 1)= ln xln z 1ln ln z 1 = ln x + ln Cln ln z 1 = ln x Cln z 1 = C x;1:4;3y:x.".#$%ln z = C x + 1 -5.C;4ln6. %: ln:;141E)$ 1 +% !5 :y= C x +1 x27.
"2. 1823.y= C x +1x2.2.4. )..$!$%H) >F%&$ ! #%$%)1, =!)#1-3 45x=u+y=v+KD. L 33 C41 ;:u, 3CDC3C45., 7 C5,7 5D:v 5a 1 x + b 1 y + c1a 2x + b2 y + c2y =f:%&0 ! #%$%)1047C5 52474= 2,=1:a 1 + b 1 + c1 = 0a 2 + b2 + c2 = 0!7 5G: 747575a 1 x + b1 y + c1 = 0, a 2 x + b 2 y + c 2 = 0 ..y =x+y 3x y 1( 243 C:::7 5+ =3=01= 0:x = u + 2 dx = duy = v + 1 dy = dvdv (u + 2) + ( v + 1) 3=du (u + 2) ( v + 1) 1dv u + v=du u v1-3 45.".2.2.5.#$%)%$'%&$ )..$!$%H) >F%&$ ! #%$%)1 1-=!$ $>$%)$ 2.6.:.
19= !1 +,:54:.1-3 45:y = a ( x ) y + b( x )4 y( x )x = a ( y) x + b ( y )4 x ( y)=!$ $>$%)$ 2.7.,b( x ) = 0 ,5Db( x ) 0 ,.;1-3 4(35:3D )&:C(4.24';145 ()@)(:4)+, )% : y = a ( x ) y + b( x )7:2;14dy= a (x ) ydx@545Ddy= a ( x ) dxy:y = C( x ) e3: y = a (x ) y - G3.ln y = a ( x )dx + ln C24%;1 3;4y = Ce33; C5 5:a ( x ) dxy = Ce,;D5a ( x ) dxa ( x ) dx; C ( x ) : y = C( x ) eC4:a ( x ) dxa ( x ) dxa ( x ) dxC (x ) e+ C( x ) a ( x ) e= a ( x ) C ( x )e+ b( x )144444424444443142434yC (x ) ea ( x ) dxC ( x ) = b ( x )e= b( x )a ( x ) dxy,4a ( x ) dx..".#$%a ( x ) dxC( x ) = b ( x ) e5: y = ( b( x ) e7J4 ::a ( x ) dxdx + Cdx + C)ea ( x ) dx2;14. 203352;.)'())1. %52.
6 4, 5 :3. ( 244.74;147.,: 422:43:: y = ( x , C) .2; 4; C:;: 4.4.,::3 C (x)C( x ) : C( x ) = C ( x )dx + C .6.7. %D39. 6 45 : y = a (x ) y . L1-3 4; C( x ) .5. %8. %.;1:54C( x ).:DC1422..2: 4. 4. L–23.".#$%. 212.4.2 y = 2x 4% : x y$G$%)$:1.:%72 ;4y;: y = 2 + 2 x 3x:: 71-3 42. 6 4 2;1:;13. ( 25 .): y = 2(42, b( x ) = 2 x 3 ,xy-Gx.:y =2yxdydx=2yxdyy=2dxxdydx=2yx4. 6a (x ) =: f ( x , y) = a ( x ) y + b( x ) , 34ln y = 2 ln x + ln C:;4;:y = C x2 ;52;:y = C( x ) x 25.4:;%3D: y = C ( x ) x 2 + C( x ) 2 x2: 4.1:2C ( x ) x 2 + C( x ) 2 x = C ( x ) x 2 + 2 x 3144424443 x 14243yyC ( x ) x 2 + 2 x C( x ) = 2 x C ( x ) + 2 x 3C ( x ) x 2 = 2x 3C ( x ) = 2x6.5; C( x ) : C( x ) = 2 x dx = x 2 + C€7. %D.8.
%29. ": y = ( x 2 + C) x 2.2€) x 2 : 4.4: y = ( x 2 + C23.".#$%. 222.5.% : y dx = ( x + y 2 )dy$G$%)$:1.4:2 ;4%C :%7:; y : y = (x + y 2 ) y:;: y =:1=yy-Gx + y21yx + y2x =x + y2yx =a ( y) =: f ( x , y) = a ( y) x + b( y ) , 31-3 42. 6 4 2;153. ( 2): x =(4x-Gy.xydxdy=xy4:;4dx x=dy ydx dy=xyln x = ln x + ln Cx=C y -;:;5;:x = C( y ) y4:;%3D: x = C ( y ) y + C( y )2: 4.1:1C ( y ) y + C( y ) = C ( y ) y + y1442443 y 123xxC ( y ) y + C( y ) = C ( y ) + yC ( y) y = yC ( y) = 16.x ( y) .:x =5.1x+yy1, b ( y) = y , : 7y5;1:4. 6.5€; C( y) : C( y) = 1 dy = y + C2.".#$%7. %D. 23€) y : 4.4: x = ( y + C223.8.
y = 02.: x = ( y + C ) y,9. "&y=0$ (01)% : y = a ( x ) y + b( x )65;5; y( x ) 4:: y ( x ) = u ( x ) v( x ) ,53 :( u v ) = a ( x ) u v + b( x )u v + u v = a ( x ) u v + b( x )C50,G 3uv+u va ( x ) u v = b( x )u v + u [va ( x ) v ] = b( x ); v( x )2dv= a (x ) vdx:;u Ceu =5a ( x ) dx,7 C;14D5 C5:ln v = a ( x )dx + ln C;lv = Ce5; u (x ) :1b ( x )eCa ( x ) dx~dx + Ca ( x ) dxdx + C)ea ( x ) dx)$5y=u v,2. %a ( x ) dxa ( x ) dx: y = u v = ( b( x )e1. %a (x ) v Cv+ u 0 = b( x )1b ( x )eCu (x ) =7C :dv= a ( x ) dxv%55, 5 :3:u v + u v = a ( x ) u v + b( x )4y =u v+u v . %.47D.".#$%u v + u [v3.-a ( x ) v] = b( x ); v , C 1 ;1 ;5G:3;1C 44.
%. 2444G7D.,15 C5 : v%7v = (x) .a(x) v = 0/4 2 .;5; v: 4 .2, 47:( x ) = b( x )uu = b( x ) / ( x )u : u = u dx + C .5.6. %D3uy=u v. Lv–2.7. %:D8. 6 442C12..2.6.2 y = 2x 4% : x y$G$%)$:1.:%72 ;4y;: y = 2 + 2 x 3x:: f ( x , y) = a ( x ) y + b( x ) , 3: 72, b( x ) = 2 x 3 ,x1-3 4y ( x ) = u ( x ) v( x ) ,2. %u v+u v =u v + [v3.5v =2; v( x ) :vx3 :2u v + 2x 3x2v =0:xvdvv=2dxxdvdx=2vxv = x2ln v = 2 ln x;5 .2v] u = 2 x 3xdvdx=2vx4. %a (x ) =5; v = x2: 4. 2.:3.".#$%.
25u x 2 = 2x 3u = 2x5.; u ( x ) : u ( x ) = 2 x dx = x 2 + C56. %v = x253u = x2 + C , 47y = ( x 2 + C) x 2 -2.7. %2.8. ": y = ( x 2 + C) x 22.2.6. )..$!$%H) >F%&$ ! #%$%)1, =!)#1E)$ 1 + >)%$'%&0.)..$!$%H) >F% $! #%$%)$ $!% >>).:=!$ $>$%)$ 2.8.D@C: 4@:y = a ( x ) y + b( x ) y n , n0,1@4 y( x )x = a ( y ) x + b ( y) x n , n0,1@4 x ( y)45z=1yn1: z( x ) =:= y1 n ,3z = (1 n ) y n yy =z yn= a ( x ) y + b( x ) y n(1 n )z yn(1 n )y= a ( x ) n + b( x )nyyz(1 n )= a ( x ) z + b( x )z = (1 n ) a ( x ) z + (1 n ) b( x ) 01.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
