Wronski-Determinantenrechner

Berechne die Wronski-Determinante von $$$\left\{f_{1} = \cos{\left(x \right)}, f_{2} = \sin{\left(x \right)}, f_{3} = \sin{\left(2 x \right)}\right\}$$$.

Die Wronski-Determinante ist durch die folgende Determinante gegeben: $$$W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}f_{1}\left(x\right) & f_{2}\left(x\right) & f_{3}\left(x\right)\\f_{1}^{\prime}\left(x\right) & f_{2}^{\prime}\left(x\right) & f_{3}^{\prime}\left(x\right)\\f_{1}^{\prime\prime}\left(x\right) & f_{2}^{\prime\prime}\left(x\right) & f_{3}^{\prime\prime}\left(x\right)\end{array}\right|.$$$

In unserem Fall gilt $$$W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\\left(\cos{\left(x \right)}\right)^{\prime } & \left(\sin{\left(x \right)}\right)^{\prime } & \left(\sin{\left(2 x \right)}\right)^{\prime }\\\left(\cos{\left(x \right)}\right)^{\prime \prime } & \left(\sin{\left(x \right)}\right)^{\prime \prime } & \left(\sin{\left(2 x \right)}\right)^{\prime \prime }\end{array}\right|.$$$

Bestimmen Sie die Ableitungen (für die Schritte siehe Ableitungsrechner): $$$W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right|.$$$

Bestimme die Determinante (für die Schritte siehe Determinantenrechner): $$$\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right| = - 3 \sin{\left(2 x \right)}.$$$

Die Wronski-Determinante ist gleich $$$- 3 \sin{\left(2 x \right)}$$$A.