Example 4. Solve the equation $x^{\prime \prime}-5 x^{\prime}+4 x=4, x(0)=0, x^{\prime}(0)=2$.
Solution. Since $4 \risingdotseq \frac{4}{p}$ and by the condition $x_{0}=x(0)=0, x_{1}=x^{\prime}(0)=2$, the operator equation will have the form $\left(p^{2}-5 p+4\right) X(p)=\frac{4}{p}+2$. From this, we find the operator solution

$$
X(p)=\frac{2 p+4}{p\left(p^{2}-5 p+4\right)}
$$

We decompose the right-hand side into partial fractions:

$$
X(p)=\frac{1}{p}-\frac{2}{p-1}+\frac{1}{p-4}
$$

Transitioning to the originals, we obtain the desired solution

$$
x(t)=1-2 e^{t}+e^{4 t} .
$$