# Trying to find laplace transform x(s) that satisfies x′(t)=x(t)−t?

Nayeli Osborne 2022-10-14 Answered
Trying to find laplace transform x(s) that satisfies ${x}^{\prime }\left(t\right)=x\left(t\right)-t$?
Taking the Laplace transform of the equation
${x}^{\prime }\left(t\right)=x\left(t\right)-t,$
we get
$sx\left(s\right)-x\left(0\right)=x\left(s\right)-\frac{1}{{s}^{2}},$
right? So if $x\left(0\right)=1$, don't you get
$x\left(s\right)=\frac{1-\frac{1}{{s}^{2}}}{s-1}?$
When I take the inverse laplace of this I get 2pi*i, how do I know this works?
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## Answers (1)

Marlene Welch
Answered 2022-10-15 Author has 23 answers
$\frac{1-\frac{1}{{s}^{2}}}{s-1}=\frac{{s}^{2}-1}{{s}^{2}\left(s-1\right)}=\frac{s+1}{{s}^{2}}=\frac{1}{s}+\frac{1}{{s}^{2}}.$
How did you get zero for the inverse Laplace transform?
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