Example 2. Find the original $f(t)$, if $F(p)=\frac{p+2}{p^{3}(p-1)^{2}}$.
Solution. The fraction $F(p)$ has a pole $p_{1}=0$ of multiplicity $n_{1}=3$ and a pole $p_{2}=1$ of multiplicity $n_{2}=2$. Using formula (16), we obtain the original

$$
\begin{aligned}
f(t)= & \frac{1}{2} \lim _{p \rightarrow 0} \frac{d^{2}}{d p^{2}}\left[\frac{p+2}{p^{3}(p-1)^{2}} p^{3} e^{p t}\right]+\lim _{p \rightarrow 1} \frac{d}{d p}\left[\frac{p+2}{p^{3}(p-1)^{2}}(p-1)^{2} e^{p t}\right]= \\
= & \frac{1}{2} \lim _{p \rightarrow 0} \frac{d^{2}}{d p^{2}}\left[\frac{p+2}{(p-1)^{2}} e^{p t}\right]+\lim _{p \rightarrow 1} \frac{d}{d p}\left(\frac{p+2}{p^{3}} e^{p t}\right)= \\
= & \frac{1}{2} \lim _{p \rightarrow 0}\left\{\left[\frac{2 p+16}{(p-1)^{4}}-\frac{2 t(p+5)}{(p-1)^{3}}+\frac{t^{2}(p+2)}{(p-1)^{2}}\right] e^{p t}\right\}+ \\
& +\lim _{p \rightarrow 1}\left\{\left[\frac{t(p+2)}{p^{3}}-\frac{3 p^{2}+5 p}{p^{4}}\right] e^{p t}\right\}=8+5 t+t^{2}+(3 t-8) e^{t}
\end{aligned}
$$

## $2^{\circ}$. Solution of the Cauchy problem for linear differential

 equations with constant coefficients. Suppose we need to find the solution of a second-order differential equation with constant coefficients

$$
x^{\prime \prime}(t)+a_{1} x^{\prime}(t)+a_{2} x(t)=f(t)
$$

satisfying the initial conditions

$$
x(0)=x_{0}, \quad x^{\prime}(0)=x_{1}
$$

We will assume that the function $f(t)$ and the solution $x(t)$, along with its derivatives up to the second order, are original functions. Let $x(t) \risingdotseq X(p), f(t) \risingdotseq F(p)$. By the rule of differentiation of originals, taking into account (2), we have

$$
x^{\prime}(t)=p X(p)-x_{0}, \quad x^{\prime \prime}(t)=p^{2} X(p)-p x_{0_{-}}-x_{1}
$$

Applying the Laplace transform to both sides of (1) and using the linearity property of the transform, we obtain the operator equation

$$
\left(p^{2}+a_{1} p+a_{2}\right) X(p)=F(p)+x_{0}\left(p+a_{1}\right)+x_{1}
$$

Solving equation (20), we find the operator solution

$$
X(p)=\frac{F(p)+x_{0}\left(p+a_{1}\right)+x_{1}}{p^{2}+a_{1} p+a_{2}}
$$

Finding the original for $X(p)$, we obtain the solution of equation (18) satisfying the initial conditions (19).

Similarly, we can solve any $n$-th order equation with constant coefficients and initial conditions at $t=0$.