Aufgabe 7.1 – Differentialgleichungssysteme aus DGL höherer Ordnung

Überführen Sie die folgenden DGL bzw. DGL-Systeme in DGL-Systeme erster Ordnung.

$y^{^{\prime\prime}} \left( x \right)+a_1 \left( x \right)y ^{\prime}\left( x \right)+a_0 \left( x \right)y\left( x \right) = b\left( x \right)$
$Ri ^{\prime}\left( t \right)+Li^{^{\prime\prime}} \left( t \right)+\frac{1} {C}i\left( t \right) = -U_0 \omega \cdot \sin \left( {\omega t} \right),\quad \quad R,L,C,U_0 ,\omega > 0$
Das System zweiter Ordnung:

$x^{^{\prime\prime}} \left( t \right) = f_1 \left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right)$

$y^{^{\prime\prime}} \left( t \right) = f_2 \left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right)$

$z^{^{\prime\prime}} \left( t \right) = f_3 \left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right)$
$F\left( {x,y\left( x \right),y ^{\prime} \left( x \right),y^{^{\prime\prime}} \left( x \right)} \right) = 0$

Lösung

a )

$y^{^{\prime\prime}} = -a_1 y ^{\prime} -a_0 y+b$

$\vec z = \left( {\begin{array}{*{20}{c}} z_1 \\ {z_2 } \\ \end{array} } \right)$

$z_1 = y$

$z_2 = y ^{\prime} = z_1 ^{\prime}$

$z_2 ^{\prime} = y^{^{\prime\prime}} = -a_1 z_2 -a_0 z_1 +b$

$\vec z ^{\prime} = \left( {\begin{array}{*{20}{c}} z_1 ^{\prime} \\ {z_2 ^{\prime} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}{c}} 0 & 1 \\ {-a_0 } & {-a_1 } \\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} z_1 \\ {z_2 } \\ \end{array} } \right)+\left( {\begin{array}{*{20}{c}} 0 \\ b \\ \end{array} } \right)$

b )

$i^{^{\prime\prime}} = -\frac{R} {L}i ^{\prime} -\frac{1} {{LC}}i-\frac{{U_0 }} {L}\omega \sin \left( {\omega t} \right)$

$\vec z = \left( {\begin{array}{*{20}{c}} z_1 \\ {z_2 } \\ \end{array} } \right)$

$z_1 = i$

$z_2 = i ^{\prime} = z_1 ^{\prime}$

$z_2 ^{\prime} = i^{^{\prime\prime}} = -\frac{R} {L}z_2 -\frac{1} {{LC}}z_1 -\frac{{U_0 }} {L}\omega \sin \left( {\omega t} \right)$

$\vec z ^{\prime} = \left( {\begin{array}{*{20}{c}} z_1 ^{\prime} \\ {z_2 ^{\prime} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}{c}} 0 & 1 \\ {\frac{1} {{LC}}} & {-\frac{R} {L}} \\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} z_1 \\ {z_2 } \\ \end{array} } \right)+\left( {\begin{array}{*{20}{c}} 0 \\ {-\frac{{U_0 }} {L}\omega \sin \left( {\omega t} \right)} \\ \end{array} } \right)$

c )

$w_1 = x$

$w_2 = x ^{\prime} = w_1 ^{\prime}$

$w_3 = y$

$w_4 = y ^{\prime} = w_3 ^{\prime}$

$w_5 = z$

$w_6 = z ^{\prime} = w_4 ^{\prime}$

$\vec w ^{\prime} = \left( {\begin{array}{*{20}{c}} w_1 ^{\prime} \\ {w_2 ^{\prime} } \\ {w_3 ^{\prime} } \\ {w_4 ^{\prime} } \\ {w_5 ^{\prime} } \\ {w_6 ^{\prime} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}{c}} & 1 & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} \\ {} & {} & {} & 1 & {} & {} \\ {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & 1 \\ {} & {} & {} & {} & {} & {} \\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} w_1 \\ {w_2 } \\ {w_3 } \\ {w_4 } \\ {w_5 } \\ {w_6 } \\ \end{array} } \right)+\left( {\begin{array}{*{20}{c}} \\ {f_1 \left( {w_1 ,w_3 ,w_5 } \right)} \\ {} \\ {f_2 \left( {w_1 ,w_3 ,w_5 } \right)} \\ {} \\ {f_3 \left( {w_1 ,w_3 ,w_5 } \right)} \\ \end{array} } \right)$