Example 4. Find the general solution of the equation

$$
y^{\mathrm{V}}-2 y^{\mathrm{IV}}+2 y^{\prime \prime \prime}-4 y^{\prime \prime}+y^{\prime}-2 y=0
$$
Solution. The characteristic equation

$$
\lambda^{5}-2 \lambda^{4}+2 \lambda^{3}-4 \lambda^{2}+\lambda-2=0
$$

or

$$
(\lambda-2)\left(\lambda^{2}+1\right)^{2}=0
$$

has roots $\lambda=2$ - a single root and $\lambda= \pm i$ - a pair of double imaginary roots. The general solution is

$$
y_{0.0}=C_{1} e^{2 x}+\left(C_{2}+C_{3} x\right) \cos x+\left(C_{4}+C_{5} x\right) \sin x
$$

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