Verifying inverse Laplace transformation of an expression

Question

I tried solving the next inverse Laplace transformation myself:

$$f(t)=L^{-1}\Bigl({{s}\over {s-C}}X(s)\Bigl)={dx(t)\over{dt}}*e^{Ct}$$

but I am not sure if it is correct. I can not find a similar example in my text books. Can anyone please tell me if this is correct?

Note: the operator $*$ is the convolution operator.

Thank you for your time.

Answer 1

It should be

$$f(t)=\mathcal{L}^{-1}\Bigl({{s}\over {s-C}}X(s)\Bigl)=\mathcal{L}^{-1}\Bigl({{s-C+C}\over {s-C}}X(s)\Bigl)=\mathcal{L}^{-1}\Bigl(X(s)+\frac{C}{s-C}X(s)\Bigl) \\ =x(t)+C x(t)*e^{Ct}$$

since $\mathcal{L}^{-1}(\frac{1}{s-C})=e^{Ct}$ and the convolution property

The slight difference is because of the extra term in the Laplace transform of the derivative

$$\mathcal{L}\left({dx(t)\over{dt}}*e^{Ct}\right)=(sX(s) \color{red}{-x(0)})\frac{1}{s-C}=\frac{s}{s-C}X(s)\color{red}{-\frac{x(0)}{s-C}}$$