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x\"+x\'-12x=t, x(0)=-1, x\'(0)=0 solve using Laplace Transform

Solution

x\"+x\'-12x=t, x(0)=-1, x\'(0)=0

taking laplace tr on both sides

s²x(s)-x(0)+sx(s)-x(0)-12x(s)=1/s²

x(s)=1/s²(s²+s-10)

taking inverce laplace on both sides gives the value of x(t)