Variacion de parametros

Variación de parámetros

1. y´´+y=secx
m2+1=0
m2=-1
m=+1i
yc=e0xC1cosx+C2senx
yc(x)=C1cosx+C2senx
W=cosxsenx-senxcosx=cos2x+sen2x=1
W1=0senxsecxcosx=-senx.secx= -senx∙1cosx=-tanx
W2=cosx0-senxsecx=cosx.secx=cosx∙1cosx=1
U´1=--tanx1
U1=-tanxdx=-senxcosxdx=duu=ln(u)=ln⁡(cosx) u = cosx du =-senx
U´2=11=1
U2=dx= x
yp(x)=lncosx.cosx+x.senxy(x)=C1cosx+C2senx-cosx∙lncosx+xsenx

2. y´´+y=tanx
m2+1=0
m2=-1
m=+1i
yc(x)=e0xC1cosx+C2senx
yc=C1cosx+C2senx
W=cosxsenx-senxcosx=cos2x+sen2x=1
W1=0senxtanxcosx=-tanx.senx=-senx.senxcosx=-sen2xcosx
W1=-1+cos2xcosx= -1cosx+ cos2xcosx= -secx+cosx
W2=cosx0-senxtanx=cosx.senxcosx=senx
U´1=-secx+cosx1=-secx+cosx
U1=-secxdx+cosxdx=-lnsecx+tanx+senx
U´2=senx1=senx
U2=senxdx= -cosxypx=-lnsecx+tanx+senxcosx-cosx ∙senx
y(x)=C1cosx+C2senx+-lnsecx+tanx+senxcosx-cosx ∙senx

3. y´´+y=senx
m2+1=0
m2=-1
m=+1i
yc(x)=e0xC1cosx+C2senx
yc(x)=C1cosx+C2senx
W=cosxsenx-senxcosx=cos2x+sen2x=1
W1=0senxsenxcosx=-sen2x
W2=cosx0-senxsenx=cosx.senx
U´1=-sen2x1= -sen2x
U´2=cosx.senx1=cosx.senx
U1=-sen2x dx=-12x+14sen2x=12senxcosx-12x
U2=cosx.senx dx=udu=u22=12sen2x u = senx du =cosxypx=cosx12senxcosx-12x+ 12sen3x
yg(x)=C1cosx+C2senx-cosx12senxcosx-12x+ 12sen3x

4. y´´+y=secθtanθ
m2+1=0
m2=-1
m=+1i
yc(θ)=e0xC1cosx+C2senx
yc(θ)=C1cosx+C2senx
W=cosxsenx-senxcosx=cos2x+sen2x=1
W1=0senxsecθtanθcosx=-senx.secθtanθ=-senx∙1cosx ∙tanx=-tan2x
W2=cosx0-senxsecθtanθ=cosx.secθtanθ=cosx∙1cosx ∙tanx=tanx
U´1=-tan2x1= tan2x
U´2=cosx.senθtanθ1=cosx.senθtanθU1=-tan2x dx=x-tanx
U2=tanθ dx=ln|cosx|
yp(θ)=cosxx-tanx+senx∙ln|cosx|
y(θ)=C1cosx+C2senx-cosxx-tanx+senx∙ln|cosx|

5. y´´+y=cos2x
m2+1=0
m2=-1
m=+1i
yc(x)=e0xC1cosx+C2senx
yc(x)=C1cosx+C2senx
W=cosxsenx-senxcosx=cos2x+sen2x=1
W1=0senxcos2xcosx=-cos2x.senx
W2=cosx0-senxcos2x=cos3x
U´1=-cos2x.senx1
U1=-cos2x.senx dx=-u2du=u33=13cos3x u = cosx du = -senxU´2=cos3x1=cos3x
U2=cos3x dx=cosx(1-sen2x)dx=cosx dx-cosx∙sen2x dx
U2=senx-u2 du=senx-u33=senx-13sen3x u = senx du = cosx
ypx=13cos3xcosx+senx(senx-13sen3x)
y(x)=C1cosx+C2senx+13cos3xcosx+senx(senx-13sen3x)

6. y´´+y=sec2x
m2+1=0
m2=-1
m=+1i
yc(x)=e0xC1cosx+C2senx
yc(x)=C1cosx+C2senx
W=cosxsenx-senxcosx=cos2x+sen2x=1
W1=0senxsen2xcosx=-sec2x.senxW2=cosx0-senxsen2x=cosx.sec2x=cosx.1cos2x=1cosx=secx
U´1=-senx.sec2x1= -senx.sec2x
U1=-senx.sec2x dx
U´2=secx1=secx
U2=secx dx=ln|secx+tanx|
yp(x)=-senx.sec2x dx .cosx+ senxln|secx+tanx|
yx=C1cosx+C2senx-senx.sec2x dx .cosx+senxlnsecx+tanx

7. y´´-y=cosh función hiperbólica cosh(x) = 12ex+e-x
m2-1=0
m2=1
m=+1
yc(x)=C1ex+C2e*x
W=exe-xex-e-x=-1-1=-2
W1=0e-xcoshx-e-x=-coshx.e-xW2=ex0excoshx=ex.coshx
U´1=-coshx.e-x-2=coshx.e-x2=12ex+e-xe-x2=12ex+12e-xe-x2=14+14e-2x
U1=14dx+14e-2xdx=14x-18e-2x
U´2=ex.coshx-2=-ex2∙coshx=12ex+e-x∙-e-x2=12ex+12e-x∙-e-x2=-14-14e-2x
U2=-14dx-14e-2xdx=-14x-18e-2x
yp(x)=14x-18e-2x .ex+-14x-18e-2x∙e-x
y(x)=C1ex+C2Xex+14x-18e-2x .ex+-14x-18e-2x∙e-x

8. y´´-y=senh2x
m2-1=0
m=±1
yc(x)=C1ex+ C2e-x
W=exe-xex-e-x=-2
W1=0e-xsenh2x-e-x=-e-xsinh2xW2=ex0exsenh2x=ex.senh2x
U´1=-xex.senh2x-2=e-x.senh2x2=e-x212e2x-12e-2x=14ex-14e-3x
U1=14exdx-14e-3xdx=14ex+112e-3x
U´2=ex.senh2x-2=-ex.senh2x2=-e-x212e2x-12e-2x=-14e3x+14e-x
U2=-14e3xdx+14e-xdx=-112e3x+14e-x
yp(x)=14ex+112e-3x .ex+-112e3x+14e-x dx .e-x
y(x)=C1ex+C2e-x+14ex+112e-3x .ex+-112e3x+14e-x dx .e-x

9. y´´-4y=e2xx
m2-4=0
m2=±4
m=±2
yc(x)=C1e2x+ C2e-2xW=e2xe-2x2e2x-2e-2x=-2-2=-4
W1=0e-2xe2xx-2e-2x=1x
W2=e2x02e2xe2xx=e4xx
U´1=-e2x.e2xx-4=-14-41=14x

U1=14xdx=14ln|x|
U´2=e2x.e2xx-4=e4xx-41=-e4x4x
U2=e4x4x dx
yp(x)=14ln|x|.e2x-14e4xx dx.e-2x
y(x)=C1e2x+ C2e2x+14ln|x|.e2x-14e4xx.e-2x

10. y´´-9y=9xe3x
m2-9=0
m2=9
m=+3
yc(x)=C1e3x+C2Xe3x
W=e3xxe3x3e3x3e3x+e3x=-3-3=-6
W1=0xe3x9xe3x3e3x+e3x=-9xe3x xe-3x= -9x2∙e-6x...