Tarea

2.3 ecuaciones exactas
1. ( x+2y+3 )dx +(2x+4y-1)dy =0
m/a =1/2 m= a/b = 1/2 si son paralelas

z= x+2y
dz=dx+2dy
dz-2dy=(z+3)(dz-2dy)+(2z-s)dy=0
Zdz+3dz-2zdy-6dy+2zdy-dy
Zdz+3dz-7dy=0
(z+3)dz -7dy=0
zdz+3dz-7dy=0
{ z22 + 3z- 7y=c } 2
z2 + 6 z -14y= c
(x+2y)2 + 6 (x+2y) -14y = c
x2+4xy+4y2+6x+12y-a4y=c
x2 +4xy+4y2+6x=c+2y

2. (x+y-3)dx+(x+y+4)dy=0
m= -1/1=-1 m=-1/1si son paralelas
z=x+y
dz=dx+dy
dx=dz-dy
(z-3)(dz-dy)+(z+4)dy=0
Zdz-3dz-zdy+3dy+zdy+4dy=0
Zdz-3dz+7dy=0
z dz-3dz+7dy=0
(z2 /2 -3z+7y=c)2
z2 -6z+14y=c
(x+y)2-6(x+y)+14y=c
x2+2xy+y2-6x-6y+14y=c
x2+2xy+y2+8y=c+6x

3. (2x-3y+2)dx +3(4x-6y-1) dy=0
M= -2/-3=2/3 m= -4/-6=4/6 si son paralelas
Z=2x-3y
Dz=2dx-3dy
dx= dz+3dy/2

{zdz+2dz+6zdy+6dy+6zdy-3dy=0}2 /2Zdz+2dz+6zdy+6dy+12zdy-6dy=0
Zdz+2dz+18zdy=0
{(z+2)dz+18zdy=0)}1/2
Z+2/2 dz+18dy= dz+2dzz+18dy
2x+15y+ln(2x-3y)2 = c

4. (zx+y)dx –( 4x+2y-1) dy =0
Zdx-(2z-1)(dz-2dx)=0
Zdx-2zdz+dz+4zdx-2dx =0
5zdx-2zdz+dz-2dx = 0} 1/z-2
-2z+1 /5z-2)dz +dx=0
-zz dz/5z-2 +dz/5z-2 +dx=0
U= 5z-2 du=z
2(5z-2)-1 dz
-2{(5z-2)z2/2 -1/10z2 du}
-{ 5z3-2z-1/5 z3/3 }
-(5z3+2z+z3/15 + 1/5 ln (5z-2)+x= c ) 1596(2x+y)3 + 2(2x+y0+ln ( 5z -2)3 +15x = c

5. (X-2Y+5)DX – { 2(X-2Y)+9}DY=0
Z=x-2y
dz=dx-2dy
dx=dz+2dy
(z+5)(dz+2dy) – ( 2z+4) dy = 0
Zdz+5dz + 2zdy +10dy -2zdy -9dy= 0
zdz+5dz+dy=0
(z2/2 +5z + y = c ) 2
(x+2y)2 +10(x+2y) 12y = c
X2-4xy+4y2+10x-20y+2y=c
X2+4y2+10x=c+4xy+18y
6. zdx +( 2x+3y ) dy =0
Z=2x+3y
dz=2dx +3dy
dx =dz-3dy/2

2(dz-3dy)/2 + zdy =0
dz-3dy+zdy=0
(dz+(z-3)dy =0) 1/z-3
dz/z-3 +dy=0
du/u +dy=0
eln(z-3)+ey = ec
(zx+3y-3)+ey = c

7. ( x+y) dx + ( x+y -2 ) dy = 0
Z ( dz-dy)+(z-2)dy = 0
Zdz –zdy +zdy -2 dy =0
z2/2 -2y = c ) 2
z2-4y = c
(x+y)2=c +4y

8. ( 3x-y+4) dx +dy =0
Z= 3x-y
Dz=3dx-dz
Dy= 3dx-dz
(z+4)dx+3dx-dz =0
Zdx +4dx + 3dx – dz = 0
Zdx +4dx +3dx –dx = 0
7dx +zdx –dz = 0
{ ( z+7) dx – dz = 0 } 1 / z+7
dx-dz/z+7 =0
dx-duu=0ex- eln( z+7) = ec
ex- ( z+7) = c
ex – 3 x + y -7 = c

9. ( 2x+3y -1 ) dx = ( 5-2x-3y ) dy
( 2x+3y -1) dx – ( -2x -3y +5 ) dy = 0
z=2x+3y
dz= 2dx+3dy
dy= dx-2dx / 3
( z-1) dx – ( -z+5) ( dz -2dx ) / 3 = 0
( zdx –dx – ( -zdz+5dz+2zdx-10dx)=0}3
3zdx-3dx+zdz-5dz-2zdx+10dx=0
Zdx+7dx+zdz-5dz=0
{(z+7)dx+(z-5)dz=0}1/z+7
dx+z-5z+7dz=0
dx+z-5 / u du =0

Ejercicios 2.51.(2xcosy+3x2y)dx + (x3-x2seny-y) dy = 0

∂M∂y= 2x(-seny)+3x2 ∂N∂x= 3x2 –seny ( 2x)

x3dy – x 2 ydy
X3dy - x2 senydy -ydy
X3y + x2cos y – y2/2

2x cosy dx+3x2ydx
2cosyxdx+3y x2ydx
2cosy ( x2/2) + 3y (x3/3)

2x2cosy + 2x3y – y 2 = c

2. ( x+seny) dx + (xcosy –zy ) dy = 0
∂M∂(Y) = cosy ∂N∂x = cosy
xdx+ seny dx
xdx+seny dx
Y2/ 2 +xseny

xcosy dy-zydy
xcosydy-z ydy
xseny – zy2/2
(x2/29xseny – y2= c ) 2
{ x2+2xseny -2 y2 = c }

3. dy/dx = 2+ye(xy)/ 2y-xe(xy) ( 2+ye(XY)) dy = 0

∂M∂y = ye(XY)(x)
zdx+ye(XY)dx
2dx+yeXYdx
2x + yexy

∂N∂x = -x exy y
2y dy- xexy dy
2ydy-xexydy

{ 2x + e xy+y2 = c }

4. dy / dx = -3yx2/ x3 +2y4 ( -3yx2) dx – ( x3 + 2y4) dy = 0

∂M∂y = 3x2∂N∂x = 3x2
-3yx2 dx
-3yx2 dx
-3y x3/3
-x34

-(x3dy + zy4dy)
-(x3dy+ zy4 dy)
-(x3y + z y5/5 )
(-x3y -2y5/5 = c ) 5
-5x3y -2y5 = c

5: 2xydx + ( 1+x2)dy = 0
∂N∂y = 2x ∂N∂x= 2x
2xy dx dy+x2dy
2yx2/2 dy+x2
X2y x2y + y = cY ( x2+1) = c

6. ( ysenx+xycosx) dx + ( xsenx+1)dy = 0

∂M∂(y)= senx + xcox ∂N∂x = xcox + senx
ysenx dx+ xycosx dx
ysenx dx+yxcosx dx
-xcosx - xcosx dx
Y ( -xcosx – senx )
-y cos x -xycosx – ysenx

xsenx dy +dy
xsenxdy+dy
xysenx + y
xysenx + y

{ xy senx + y = x }

7. { e2y – y cos ( xy ) + 2y } + { 2xe2y –xcos ( xy ) + 2y + 2 x } dy =...