Example. Restore the original from its image

$$
\frac{1}{\left(p^{2}+1\right)^{2}}
$$
Solution. In this case

$$
F(p)=G(p)=\frac{1}{p^{2}+1}
$$

1. We restore the original $f(t)$ from its image $F(p)$:

$$
\sin t \longleftrightarrow \frac{1}{p^{2}+1}
$$

By the convolution theorem

$$
\sin t * \sin t \longleftrightarrow \frac{1}{p^{2}+1} \cdot \frac{1}{p^{2}+1}
$$

## 2. We compute the convolution

$$
\sin t * \sin t=\int_{0}^{\tau} \sin (t-\tau) \sin \tau d \tau=\frac{1}{2} t \cos t-\frac{1}{2} \sin t
$$

3. By formula (1), we write the answer:

$$
f(t)=\frac{1}{2} t \cos t-\frac{1}{2} \sin t
$$

Answer. $f(t)=\frac{1}{2} t \cos t-\frac{1}{2} \sin t$.

Conditions of the Problems. Restore the original by the image $F(p)$, using the convolution theorem.

1. $F(p)=\frac{4}{\left(p^{2}+4\right)^{2}}$.
2. $F(p)=\frac{9}{\left(p^{2}+9\right)^{2}}$.
3. $F(p)=\frac{1}{p^{2}\left(p^{2}-1\right)^{2}}$.
4. $F(p)=\frac{p^{2}}{\left(p^{2}+4\right)\left(p^{2}+9\right)}$.
5. $F(p)=\frac{p}{p^{4}-1}$.
6. $F(p)=\frac{p^{2}}{\left(p^{2}+1\right)^{2}}$.
7. $F(p)=\frac{e^{-p}}{p\left(p^{2}+1\right)}$.
8. $F(p)=\frac{e^{-p}}{p^{2}\left(p^{2}+1\right)}$.
9. $F(p)=\frac{e^{-p}}{p\left(p^{2}+1\right)^{2}}$.
10. $F(p)=\frac{1-e^{-p}}{p\left(p^{2}+1\right)}$.

Answers. 1. $f(t)=(\sin 2 t-2 t \cos 2 t) / 4$. 2. $f(t)=(\sin 3 t-3 t \cos 3 t) / 6$.
3. $f(t)=(t \cosh t-3 t \sinh t+2 t) / 2$. 4. $f(t)=(3 \sin 3 t-2 \sin 2 t) / 5$.
5. $f(t)=(\cosh t-\cos t) / 2$. 6. $f(t)=(\sin t+t \cos 3 t) / 2$.
7. $f(t)=(1-\cos t) \eta(t-1)$. 8. $f(t)=(t-1-\sin (t-1)) \eta(t-1)$.
9. $f(t)=(1-\cos (t-1)-(t-1) \sin (t-1) / 2) \eta(t-1)$.
10. $f(t)=(1-\cos t) \eta(1-t)+(\cos (t-1)-\cos t) \eta(t-1)$.

### 2.11. Restoration of the Original

 from the image $R(p) e^{-p \tau}$Problem Statement. Restore the original $f(t)$ from its image

$$
F(p)=R(p) e^{-p \tau}
$$

where $R(p)$ is a proper rational fraction and $\tau>0$.

SOLUTION PLAN.

1. Restore the original $r(t)$ from its image $R(p)$.
2. By the delay theorem, the desired original is given by the formula

$$
f(t)=r(t-\tau) \eta(t-\tau)
$$