Aufgabe 7.1 – Differentialgleichungssysteme aus DGL höherer Ordnung

 

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

  1. 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)

  2. 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

  3. 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)

  4. 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)

d )

w_1  = y

w_2  = y ^{\prime}   = w_1 ^{\prime}

F\left( {x,w_1 ,w_2 ,w_2 ^{\prime}  } \right) = 0

G\underbrace {\left( {w_1 ^{\prime}  ,w_2 } \right)}_{x,w_1 ,w_1 ^{\prime}  ,w_2 ,w_2 ^{\prime}  } = w_1 ^{\prime}  -w_2  = 0