Question

7. In Problems 1-30 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.

${\mathcal{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{s^2}-\frac{1}{s}+\frac{1}{s-2}\right\}$

Short answer

$\mathrm{t}-1+{\mathrm{e}}^{2\mathrm{t}}$

Step-by-step solution

Definition of inverse Transform

Consider the formula for inverse Laplace given below:

1. $1={\mathcal{L}}^{-1}\left\{\frac{1}{\mathrm{s}}\right\}$
2. ${\mathrm{t}}^{\mathrm{n}}={\mathcal{L}}^{-1}\left\{\frac{\mathrm{n}!}{{\mathrm{s}}^{\mathrm{n}+1}}\right\},\mathrm{n}=1,2,3,\dots$
3. ${\mathrm{e}}^{\mathrm{at}}={\mathcal{L}}^{-1}\left\{\frac{1}{\mathrm{s}-\mathrm{a}}\right\}$
4. $\mathrm{sinkt}={\mathcal{L}}^{-1}\left\{\frac{\mathrm{k}}{{\mathrm{s}}^2+{\mathrm{k}}^2}\right\}$
5. $\mathrm{coskt}={\mathcal{L}}^{-1}\left\{\frac{\mathrm{s}}{{\mathrm{s}}^2+{\mathrm{k}}^2}\right\}$
6. $\mathrm{sinhkt}={\mathcal{L}}^{-1}\left\{\frac{\mathrm{k}}{{\mathrm{s}}^2-{\mathrm{k}}^2}\right\}$
7. $\mathrm{coshkt}={\mathcal{L}}^{-1}\left\{\frac{\mathrm{s}}{{\mathrm{s}}^2-{\mathrm{k}}^2}\right\}$
   

Determine the inverse Laplace transform:

Resolve the expression as:

${\mathrm{L}}^{-1}\left\{\frac{1}{{\mathrm{s}}^2}-\frac{1}{\mathrm{s}}+\frac{1}{\mathrm{s}-2}\right\}={\mathrm{L}}^{-1}\left\{\frac{1}{{\mathrm{s}}^2}\right\}-{\mathrm{L}}^{-1}\left\{\frac{1}{\mathrm{s}}\right\}+{\mathrm{L}}^{-1}\left\{\frac{1}{\mathrm{s}-2}\right\}$

From the formulas, solve for the inverse Laplace as:

${\mathrm{L}}^{-1}\left\{\frac{1}{{\mathrm{s}}^2}-\frac{1}{\mathrm{s}}+\frac{1}{\mathrm{s}-2}\right\}=\mathrm{t}-1+{\mathrm{e}}^{2\mathrm{t}}$