Example 1. Find the general solution of the differential equation

$$
y^{\prime \prime \prime}-4 y^{\prime \prime}+7 y^{\prime}-4 y=0
$$
Solution. The roots of the characteristic equation $k^{3}-4 k^{2}+$ $+7 k-4=0$ can be determined by the method of factorization: $\left(k^{2}-3 k+4\right)(k-1)=0$.

We find $k_{1}=1, k_{2}=\frac{3-\sqrt{7} i}{2}, k_{2}=\frac{3+\sqrt{7} i}{2}$.

Answer. $y_{00}=C_{1} e^{x}+\left(C_{2} \cos \frac{\sqrt{7}}{2} x+C_{3} \sin \frac{\sqrt{7}}{2} x\right) e^{(3 / 2) x}$.