1.
k2+1=0
k=±i
yh=c1cosx+c2sinx
yp=Axcosx+Bxsinx
yp′=Acosx−Axsinx+Bsinx+Bxcosx=(A+Bx)cosx+(B−Ax)sinx
yp′′=Bcosx−(A+Bx)sinx+(B−Ax)cosx−Asinx
Bcosx−(A+Bx)sinx+(B−Ax)cosx−Asinx+Axcosx+Bxsinx=sinx
B=0
−A−A=1
A=−1/2
y=yh+yp=c1cosx+c2sinx−xcosx/2
2.
k2+4k+5=0
k=2−4±16−20=−2±2i
yh=e−2x(c1cos2x+c2sin2x)
yp1=Ax+B
4A+5Ax+5B=50x
A=10,B=−40/5=−8
yp1=10x−8
yp2=Ae3x
9Ae3x+12e3x+5e3x=13e3x
A=−4/9
yp2=−4e3x/9
y=e−2x(c1cos2x+c2sin2x)+10x−8−4e3x/9
3.
k3+k2−4k−4=0
k(k2−4)+k2−4=0
k1=−1,k2=−2,k3=2
yh=c1e−x+c1e−2x+c1e2x
yp=Acosx+Bsinx
yp′=Bcosx−Asinx
yp′′=−Bsinx−Acosx
yp′′′=−Bcosx+Asinx
−Bcosx+Asinx−Bsinx−Acosx−4(Bcosx−Asinx)−4(Acosx+Bsinx)=
=4sinx
−5B−5A=0
5A−5B=4
10A=4⟹A=2/5,B=−2/5
y=c1e−x+c1e−2x+c1e2x+2(cosx−sinx)/5
4.
k3−1=0
(k−1)(k2+k+1)=0
k1=1
k2+k+1=0
k2,3=2−1±i3
yh=c1ex+e−x/2(c2cos(x3/2)+c3sin(x3/2))
yp=Ax+B
−Ax−B=x
A=−1,B=0
y=c1ex+e−x/2(c2cos(x3/2)+c3sin(x3/2))−x
5.
k2−4k+4=0
k1,2=2
yh=c1e2x+c2xe2x
yp=Aex
A−4A+4A=1
A=1
y=c1e2x+c2xe2x+ex
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