a)
yk−yk−1+2yk−2=k²+5k
charasteristic equation:
r2−r+2=0
r=21±i7
yh=rk(acos(kθ)+bsin(kθ))
r=(1/2)2+(7/2)2=2
θ=arccos(1/(22))=1.2 rad
yh=(2)k(acos(1.2k)+bsin(1.2k))
yt=Ak2+Bk+C
Ak2+Bk+C+A(k−1)2+B(k−1)+C+2A(k−2)2+2B(k−2)+2C=
=k2+5k
4A=1⟹A=1/4
4B−10A=5⟹B=1.875
4C+9A−5B=0⟹C=1.78
yk=(2)k(acos(1.2k)+bsin(1.2k))+0.25k2+1.875k+1.78
b)
yk+2−4yk+1+yk=3k+2k
charasteristic equation:
r2−4r+1=0
r=2±3
yh=a(2−3)k+b(2+3)k
yt1=Ak+B
A(k+2)+B−4(A(k+1)+B)+Ak+B=3k
−2A=3⟹A=−1.5
−2A−2B=0⟹B=1.5
yt2=A2k
A2k+2−4A2k+1+A2k=2k
4A−8A+A=1
A=−1/3
yk=a(2−3)k+b(2+3)k−1.5k+1.5−2k/3
Comments
Leave a comment