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_N3WTON_
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Homework Statement
Division by s Equals integration by t:
For this problem use the following property (see relevant equations) to find the inverse transform of the given function: [itex] F(s) = \frac{1}{s(s-1)} [/itex]
Homework Equations
[itex] L^{-1}(\frac{F(s)}{s}) = \int_{0}^{t} f(\tau)\,d \tau [/itex]
The Attempt at a Solution
Using the above formula I have: [itex] F(s) = \frac{1}{s(s-1)} [/itex]
So : [itex] \frac{F(s)}{s} = \frac{\frac{1}{s(s-1)}}{s} = \frac{1}{s^{2}(s-1)} [/itex]
Next I used a partial fraction decomposition:
[itex] \frac{1}{s^{2}(s-1)} = \frac{A}{s} + \frac{B}{s^{2}} + \frac{C}{(s-1)} [/itex]
[itex] As(s-1) + B(s-1) + Cs^{2} = 1 [/itex]
Solving for A, B, and C I found [itex] A=-1 \hspace{1 mm} B=-1 \hspace{1 mm} C = 1 [/itex]
So the decomposition is:
[itex] \frac{1}{s^{2}(s-1)} = -\frac{1}{s} - \frac{1}{s^{2}} + \frac{1}{(s-1)} [/itex]
The inverse Laplace transform of the above equation is:
[itex] -t - e^{t} - 1 [/itex]
However, the answer given in my textbook is:
[itex] e^{t} - 1 [/itex]
Coincidentally, when I perform the partial fraction decomposition on [itex] F(s) [/itex], I arrive at the answer given in the back of the text, which is really confusing me, do I not need to divide [itex]F(s)[/itex] by [itex] s [/itex]?
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