
 \textbf{10.} �e�te homogenn� rovnice:
    \begin{align*}
        y^{(4)} + 2y^{(2)} + y  &= 0\\
        y^{(4)} + 2y''' + y''   &= 0\\
        y'' - 5y' + 6y      &= 0\\
        y'' + 4y' + 13y     &= 0
    \end{align*}

    \textbf{�e�en�}
    \begin{enumerate}
    \item $y^{(4)} + 2y^{(2)} + y = 0$. Charakteristick� polynom t�to rovnice je
    $$
        M(\lambda) = \lambda^4 + 2\lambda^2 + 1 = (\lambda^2 + 1)^2
    $$ Jeho ko�eny jsou
    $$
        \mu = \pm i \quad\text{ka�d� n�sobnosti 2},
    $$
    tedy fundament�ln� syst�m je $\lbrace \sin x, \cos x, x\sin x, x\cos x \rbrace$ a
    �e�en� rovnice jsou ve tvaru:
    $$
        y = A\sin x + B\cos x + Cx\sin x + Dx\cos x, \quad\text{kde }A,B,C,D \in \mathbb R.
    $$

    \item $y^{(4)} + 2y''' + y'' = 0$.\\
    Charakteristick� polynom: $M(\lambda) = \lambda^4 + 2\lambda^3 + \lambda^2 = \lambda^2(\lambda + 1)^2$.\\
    Ko�eny: $\lambda = 0$ --- dvojn�sobn�, $\lambda = -1$ --- dvojn�sobn�.\\
    Fundament�ln� syst�m: $\lbrace 1, x, e^{-x}, xe^{-x} \rbrace$\\
    �e�en�: $y = A + Bx + Ce^{-x} + Dxe^{-x}$, $A, B, C, D \in \mathbb R$.
    \item $y'' - 5y' + 6y = 0$.\\
    Charakteristick� polynom: $M(\lambda) = \lambda^2 - 5\lambda + 6 = (\lambda - 2)(\lambda - 3)$.\\
    Ko�eny: $\lambda = 2$ --- jednon�sobn�, $\lambda = 3$ --- jednon�sobn�.\\
    Fundament�ln� syst�m: $\lbrace e^{2x}, e^{3x} \rbrace$.\\
    �e�en�: $y = Ae^{2x} + Be^{3x}$, kde $A, B \in \mathbb R$.
    \item $y'' + 4y' + 13y = 0$.\\
    Charakteristick� polynom: $M(\lambda) = \lambda^2 + 4\lambda + 13$.\\
    Ko�eny: $\mu = -1 \pm 3i$ --- oba jednon�sobn�.\\
    Fundament�ln� syst�m: $\lbrace e^{-x}\sin 3x, e^{-x}\cos 3x\rbrace$.\\
    �e�en�: $y = e^{-x}(A\sin 3x + B\cos 3x)$, kde $A, B \in \mathbb R$.
    \end{enumerate}
