Properties of Laplace Transforms

$\begin{array}{c}\mathbf{1}\mathbf{.}\mathit{Linearity}\mathit{Property}\hfill \\ \mathit{If}\mathit{a}\mathrm{,}\mathit{b}\mathit{are}\mathit{any}\mathrm{constants}\mathit{and}\mathit{f}\mathrm{,}\mathit{g}\mathit{are}\mathit{any}\mathit{functions}\mathit{of}\mathit{t}\hfill \\ \mathit{then}\mathrm{,}\hfill \\ \mathit{L}\left[\mathit{af}\mathrm{(}\mathit{t}\mathrm{)}+\mathit{bg}\mathrm{(}\mathit{t}\mathrm{)}\right]=\mathit{aL}\left[\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\right]+\mathit{bL}\left[\mathit{g}\mathrm{(}\mathit{t}\mathrm{)}\right]\hfill \\ \hfill \\ \mathbf{2}\mathbf{.}\mathit{First}\mathit{Shifting}\mathit{Property}\hfill \\ \mathit{If}\mathit{L}\left[\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\right]=\mathit{f}\mathrm{(}\mathit{s}\mathrm{)}\mathit{then}\mathrm{,}\hfill \\ \mathit{L}\left[{\mathit{e}}^{\mathit{at}}\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\right]=\mathit{f}\mathrm{(}\mathit{s}-\mathit{a}\mathrm{)}\hfill \\ \hfill \\ \mathbf{3}\mathbf{.}\mathit{Change}\mathit{of}\mathit{Scale}\mathit{Property}\hfill \\ \mathit{If}\mathit{L}\left[\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\right]=\mathit{f}\mathrm{(}\mathit{s}\mathrm{)}\mathit{then}\mathrm{,}\hfill \\ \mathit{L}\left[\mathit{f}\mathrm{(}\mathit{at}\mathrm{)}\right]=\frac{1}{\mathit{a}}\mathit{f}\left(\frac{\mathit{s}}{\mathit{a}}\right)\hfill \end{array}$