$$$\cosh{\left(x \right)}$$$, $$$\sinh{\left(x \right)}$$$, $$$\cos{\left(x \right)}$$$, $$$\sin{\left(x \right)}$$$ のウロン行列式
入力内容
$$$\left\{f_{1} = \cosh{\left(x \right)}, f_{2} = \sinh{\left(x \right)}, f_{3} = \cos{\left(x \right)}, f_{4} = \sin{\left(x \right)}\right\}$$$ のウロンスキ行列式を計算してください。
解答
ワロンスキアンは次の行列式で与えられます: $$$W{\left(f_{1},f_{2},f_{3},f_{4} \right)}\left(x\right) = \left|\begin{array}{cccc}f_{1}\left(x\right) & f_{2}\left(x\right) & f_{3}\left(x\right) & f_{4}\left(x\right)\\f_{1}^{\prime}\left(x\right) & f_{2}^{\prime}\left(x\right) & f_{3}^{\prime}\left(x\right) & f_{4}^{\prime}\left(x\right)\\f_{1}^{\prime\prime}\left(x\right) & f_{2}^{\prime\prime}\left(x\right) & f_{3}^{\prime\prime}\left(x\right) & f_{4}^{\prime\prime}\left(x\right)\\f_{1}^{\prime\prime\prime}\left(x\right) & f_{2}^{\prime\prime\prime}\left(x\right) & f_{3}^{\prime\prime\prime}\left(x\right) & f_{4}^{\prime\prime\prime}\left(x\right)\end{array}\right|$$$
この場合、$$$W{\left(f_{1},f_{2},f_{3},f_{4} \right)}\left(x\right) = \left|\begin{array}{cccc}\cosh{\left(x \right)} & \sinh{\left(x \right)} & \cos{\left(x \right)} & \sin{\left(x \right)}\\\left(\cosh{\left(x \right)}\right)^{\prime } & \left(\sinh{\left(x \right)}\right)^{\prime } & \left(\cos{\left(x \right)}\right)^{\prime } & \left(\sin{\left(x \right)}\right)^{\prime }\\\left(\cosh{\left(x \right)}\right)^{\prime \prime } & \left(\sinh{\left(x \right)}\right)^{\prime \prime } & \left(\cos{\left(x \right)}\right)^{\prime \prime } & \left(\sin{\left(x \right)}\right)^{\prime \prime }\\\left(\cosh{\left(x \right)}\right)^{\prime \prime \prime } & \left(\sinh{\left(x \right)}\right)^{\prime \prime \prime } & \left(\cos{\left(x \right)}\right)^{\prime \prime \prime } & \left(\sin{\left(x \right)}\right)^{\prime \prime \prime }\end{array}\right|$$$ となります。
導関数を求めよ（手順は derivative calculator を参照）：$$$W{\left(f_{1},f_{2},f_{3},f_{4} \right)}\left(x\right) = \left|\begin{array}{cccc}\cosh{\left(x \right)} & \sinh{\left(x \right)} & \cos{\left(x \right)} & \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & - \sin{\left(x \right)} & \cos{\left(x \right)}\\\cosh{\left(x \right)} & \sinh{\left(x \right)} & - \cos{\left(x \right)} & - \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & \sin{\left(x \right)} & - \cos{\left(x \right)}\end{array}\right|$$$。
行列式を求めよ (手順については determinant calculator を参照): $$$\left|\begin{array}{cccc}\cosh{\left(x \right)} & \sinh{\left(x \right)} & \cos{\left(x \right)} & \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & - \sin{\left(x \right)} & \cos{\left(x \right)}\\\cosh{\left(x \right)} & \sinh{\left(x \right)} & - \cos{\left(x \right)} & - \sin{\left(x \right)}\\\sinh{\left(x \right)} & \cosh{\left(x \right)} & \sin{\left(x \right)} & - \cos{\left(x \right)}\end{array}\right| = 4$$$
解答
ウロンスキ行列式は$$$4$$$Aに等しい。