Ecuaciones diferenciales

CORPORACION UNIVERSITARIA DE LA COSTA (CUC)

ECUACIONES DIFERENCIALES
LINEALES DE PRIMER ORDEN

PROFESOR: FRANCISCO RACEDO

INTEGRANTES:
MALDONADO CHARRIS MARIA ROSA
PACHECO NAVARRO TONY

INGENIERIA INDUSTRIAL

BARRANQUILLA-ATLANTICO
2012
1)dydx=5y
dydx-5y=0
e-sdx = e-5dx= e-5x
ddxe-5x.y=o.e-5x
d e-5x.y=o.e-5xdx
e-5x.y=o
e-5x.y=c
y=ce-5x
y=ce5x
2)dydx=2y
dydx-2y=0e-2dx = e-2x
ddxe-2x.y=e-2x.0
de-2x.y=e-2x.0dx
e-2x.y=e-2x.0dx
e-2x.y=0
e-2x.y=c
y=ce-2x=ce-2x
3)dydx+y=e3xu=-2xdu=-2dx
edx=ex
ddxex.y=e-2xdx
dex.y=-12eu-du
ex.y=-12eu+c
ex.y= -12e-2x+c
y=-12e-2x+cex=-12.e-2xex
y=--12e-2x.ex+ce-x
y=-12e-3x+ce-x

4)3dydx+12y=4u=4xdu=4dx
dydx+4y=43dx=du4
e4dx=e4x
ddxe4x.y=43e4xdx
de4x.y=43e4xdx
e4x.y=43eu.du4=13eu.du
e4x.y=13eu+c
e4x.y=13e4x+cy=13+ce-4x
5)y`+3x2y=10x2u=x3du=3x2dx
e3x2dx=e3x2dx=ex3du3=x2dx
ddxex3.y=10x2ex3
dex3.y=103x2ex3dx
ex3.y=103eudu
ex3.y=103eu+c
ex3.y=103ex3+c
y=103+ce-x3
6)y`+2xy=x3u=x2 du=2xdx du2x=dx
e2x22 =ex2
ddxex2.y=ex2.x3
ex2.y=x3ex2dx
ex2.y=x3eudu2x
ex2.y=x2eudu2
12=ueudu
12=ueu-eu+c
12=x2ex2-ex2+c
ex2.y=12x2ex2-ex2+c
ex2.y=12x2ex2-ex21+c
y=x2ex22ex2.ex22ex2+c2ex2yx=12x2-12+c2e-x2
7) x2y´+xy=1
x2x2dydx+xx2y=1x2
dydx+xx2y=1x2 , p(x) = xx2 y f(X) = 1x2
ex21dx = ex2dx = elnx = x
dydx(x.y)=x1x2
ddxx.y=1x
dx.y=1xdx
x.y=lnx+c
y=x-1lnx+x-1c

8) y´=2y+ x2+5
dydx-2y= x2+5
e-2dx = e-2x
dydx-e-2x.y=x2+5.e-2x
d-e-2x.y= x2.e-2xdx+5e-2xdx

x2.e-2xdx U= x2 dv = e-2xdx
Du= 2x.dx v= -1/2e-2x
-x22e-2x+22e-2xx dx
U= x dv= e-2xdx
Du= dx v= -1/2 e-2x
-x22e-2x=x2e-2x+12e-2xdx

-x22e-2x-x2e-2x-14e-2x+c
y. e-2x=- x22e-2x-x2e-2x-14e-2x-52e-2x+c

y. e-2x=- x22e-2x-x2e-2x-114e-2x+c

y=- x22-x2-114 +ce-2x

9) xdydx-y=x2senx

dydx-yx=x2senxx

dydx-yx=xsenxp(x)= -1/x f(x) = x sen x

e-1xdx = e-lnx = x-1

ddxx-1.y=x-1.xsenx

x-1.y=senx.dxx-1.y=-cosx+c

y=cx-xcosc

10) xdydx+2y=3

dydx+2xy =3xp(x) = 2/x f(x) = 3/x

e2x dx = e2dxx = e2lnx = x2

ddxx2.y=x2.3x

x2.y= 3x.dx

x2.y=32x2+c

y=32+cx-2
11) xdydx+ 4y=x3-x

dydx+4xy=x2-1p(x) = 4/x f(x) = x2-1

e4xdx = e4dxx =e4lnx = x4

ddx(x4.y)=x4(x2-1)

d+x4.y= (x6-x4)dx

x4.y=x6dx-x4dx

x4.y=x77-x55+c

y=x37-x5+ x-4c

12) 1+x dydx-xy=x+x2dy dx-x1+xy=x+x21+xp(x) = -x1+x=1x+1-1

dy dx-x1+xy=x(1+x)1+x

dy dx-x1+xy=x

e11+x-1 .dx = edx1+x-dx = elnx+1-x = elnx+1-e-x = (x+1)-e-x

d dx-x+1-e-x.y=x+1-e-x x

d-x+1-e-x.y=x+1-e-x x dx

d-x+1-e-x.y=x+1-e-x x dx

x+1-e-x.y= x2e-xdx+ xe-xdx

x2e-xdx u= x2 dv= e-xdx
Du= 2x.dx v= -e-x
uv-vdu= -x2ex+2xe-xdx

U= x dv= e-xdx
Du= dx v=- e-x

-x2ex+2(-e-xx+ e-xdx)

-x2ex+2(-e-xx-e-x+c)

U= x dv= e-xdx
Du= dx v= -e-x

-xe-x+ e-xdx

-xe-x-e-x+c

x+1e-x.y=-x2e-x+2e-xx-2e-x-xe-x-e-x+c

x+1e-x.y=-x2e-x+3e-xx-3e-x+ce-x

x+1e-x.y=e-x(-x2 +3x-3)+ce-x

x+1.y=-x2 +3x-3+ce-x

x+1.y=-xx+1-2x-3+ce-x

y=-x -2x-3x+1+ce-xx+1

13) x2y´+xx+2y=e-x

x2x2dydx+xx+2x2y=e-xx2

dydx+1+2xy=e-xx2p(x)= 1+2x f(x)= e-xx2

edx+2dxx

ex+2lnx=ex+2lnx= x2ex

dydx(x2exy)=x2exe-xx2
dydx(x2exy)=1

d(x2exy)=dx

x2exy=x+c

y=xx2ex+cx2ex

y=xe-x+cx2e-x

14) xy´+1+xy=e-xsen2x

xxdydx+1+xxy=e-xsen2xx

dydx+(1x+1)y=e-xsen2xxp(x)= 1x+1 f(x)= e-xsen2xx

edxx+dx = elnx+x = xexddx(xexy)=xex.e-xsen2xx

dxexy=sen2x.dx

dxexy=sen2x.dx

xexy= -12cos2x+c

y= -12cos2xxe x +cxex

15) ydx-4x+y6dy=0
ydx dy =4x+4y6
ydxdy-4x=4y6 PY=-4y
Fy=4y5
dxdy-4yx=4y5
py=-4y dy

=-4dyy=-4ln(y)=elnY-4=y-4
y4.x=4y5.y-4 dy=4 y22
x=2y6+cy4
16) ydx=(y ey-2x)dy
ydxdy+2x=yey PY=2y
ydxdy+2x=yey FY=ey
dxdy+2y x= ey
PY=2y dy
2dyy=2lnY=eln(y)2=y2...