Wronskin determinantti funktioille $$$\cosh{\left(x \right)}$$$, $$$\sinh{\left(x \right)}$$$, $$$\cos{\left(x \right)}$$$, $$$\sin{\left(x \right)}$$$

Laske Wronskin determinantti funktioille $$$\left\{f_{1} = \cosh{\left(x \right)}, f_{2} = \sinh{\left(x \right)}, f_{3} = \cos{\left(x \right)}, f_{4} = \sin{\left(x \right)}\right\}$$$.

Wronskin determinantti määritellään seuraavalla determinantilla: $$$W{\left(f_{1},f_{2},f_{3},f_{4} \right)}\left(x\right) = \left|\begin{array}{cccc}f_{1}\left(x\right) & f_{2}\left(x\right) & f_{3}\left(x\right) & f_{4}\left(x\right)\\f_{1}^{\prime}\left(x\right) & f_{2}^{\prime}\left(x\right) & f_{3}^{\prime}\left(x\right) & f_{4}^{\prime}\left(x\right)\\f_{1}^{\prime\prime}\left(x\right) & f_{2}^{\prime\prime}\left(x\right) & f_{3}^{\prime\prime}\left(x\right) & f_{4}^{\prime\prime}\left(x\right)\\f_{1}^{\prime\prime\prime}\left(x\right) & f_{2}^{\prime\prime\prime}\left(x\right) & f_{3}^{\prime\prime\prime}\left(x\right) & f_{4}^{\prime\prime\prime}\left(x\right)\end{array}\right|.$$$

Meidän tapauksessamme pätee $$$W{\left(f_{1},f_{2},f_{3},f_{4} \right)}\left(x\right) = \left|\begin{array}{cccc}\cosh{\left(x \right)} & \sinh{\left(x \right)} & \cos{\left(x \right)} & \sin{\left(x \right)}\\\left(\cosh{\left(x \right)}\right)^{\prime } & \left(\sinh{\left(x \right)}\right)^{\prime } & \left(\cos{\left(x \right)}\right)^{\prime } & \left(\sin{\left(x \right)}\right)^{\prime }\\\left(\cosh{\left(x \right)}\right)^{\prime \prime } & \left(\sinh{\left(x \right)}\right)^{\prime \prime } & \left(\cos{\left(x \right)}\right)^{\prime \prime } & \left(\sin{\left(x \right)}\right)^{\prime \prime }\\\left(\cosh{\left(x \right)}\right)^{\prime \prime \prime } & \left(\sinh{\left(x \right)}\right)^{\prime \prime \prime } & \left(\cos{\left(x \right)}\right)^{\prime \prime \prime } & \left(\sin{\left(x \right)}\right)^{\prime \prime \prime }\end{array}\right|.$$$

Laske derivaatat (vaiheet: katso derivointilaskuri): $$$W{\left(f_{1},f_{2},f_{3},f_{4} \right)}\left(x\right) = \left|\begin{array}{cccc}\cosh{\left(x \right)} & \sinh{\left(x \right)} & \cos{\left(x \right)} & \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & - \sin{\left(x \right)} & \cos{\left(x \right)}\\\cosh{\left(x \right)} & \sinh{\left(x \right)} & - \cos{\left(x \right)} & - \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & \sin{\left(x \right)} & - \cos{\left(x \right)}\end{array}\right|.$$$

Laske determinantti (vaiheet: ks. determinanttilaskin): $$$\left|\begin{array}{cccc}\cosh{\left(x \right)} & \sinh{\left(x \right)} & \cos{\left(x \right)} & \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & - \sin{\left(x \right)} & \cos{\left(x \right)}\\\cosh{\left(x \right)} & \sinh{\left(x \right)} & - \cos{\left(x \right)} & - \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & \sin{\left(x \right)} & - \cos{\left(x \right)}\end{array}\right| = 4$$$

Wronskin determinantti on $$$4$$$A.