Calculadora Wronskiana

Calcular el Wronskian de $\left\{f_{1} = \cos{\left(x \right)}, f_{2} = \sin{\left(x \right)}, f_{3} = \sin{\left(2 x \right)}\right\}$.

El wronskiano viene dado por el siguiente determinante: $W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}f_{1}\left(x\right) & f_{2}\left(x\right) & f_{3}\left(x\right)\\f_{1}^{\prime}\left(x\right) & f_{2}^{\prime}\left(x\right) & f_{3}^{\prime}\left(x\right)\\f_{1}^{\prime\prime}\left(x\right) & f_{2}^{\prime\prime}\left(x\right) & f_{3}^{\prime\prime}\left(x\right)\end{array}\right|.$

En nuestro caso, $W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\\left(\cos{\left(x \right)}\right)^{\prime } & \left(\sin{\left(x \right)}\right)^{\prime } & \left(\sin{\left(2 x \right)}\right)^{\prime }\\\left(\cos{\left(x \right)}\right)^{\prime \prime } & \left(\sin{\left(x \right)}\right)^{\prime \prime } & \left(\sin{\left(2 x \right)}\right)^{\prime \prime }\end{array}\right|.$

Halla las derivadas (para ver los pasos, consulta calculadora de derivadas): $W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right|.$

Halla el determinante (para ver los pasos, consulta calculadora de determinantes): $\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right| = - 3 \sin{\left(2 x \right)}.$

El Wronskian es igual a $- 3 \sin{\left(2 x \right)}$A.