Given IVP:
8x′′(t)+16x′(t)−5x(t)=0
x(0)=16,x′(0)=−8The characteristic equation is as follows
8k2+16k−5=0Roots:
k1=−1−26/4,k1=−1+26/4General solution of IVP:
x(t)=Aek1t+Bek2t=e−t(Ae−26/4t+Be26/4t)The initial conditions give
x(0)=A+B=16x′(0)=−(A+B)+26/4(−A+B)=−8A=−8/13(−13+26),B=8/13(13+26)Finally,
x(t)=e−t(−8/13(−13+26)e−26/4t+8/13(13+26)e26/4t)
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