3.41. Using the convolution theorem, find the original function of $\bar{f}(p)=\frac{p}{p^{4}-1}$.
Solution. Let's write $\bar{f}(p)$ as

$$
\bar{f}(p)=\frac{p}{p^{2}-1} \frac{1}{p^{2}+1}
$$

Since $\frac{p}{p^{2}-1} \doteqdot \operatorname{ch} t$ (formula 8 in table 3.2 for $\lambda=1$) and $\frac{1}{p^{2}+1} \doteqdot \sin t$ (formula 5 in table 3.1 for $\omega=1$), by the convolution theorem (formula 14 in table 3.1), we have:

$$
\frac{p}{p^{4}-1} \doteqdot \int_{0}^{t} \operatorname{ch}(t-\tau) \sin \tau d \tau=\frac{\operatorname{ch} t-\cos t}{2}
$$

### 3.14 .8 .

## APPLICATION

OF OPERATIONAL CALCULUS TO THE SOLUTION OF SOME DIFFERENTIAL EQUATIONS AND SYSTEMS OF DIFFERENTIAL EQUATIONS