Wronskian dari $$$x$$$, $$$x^{3}$$$

Hitung Wronskian dari $$$\left\{f_{1} = x, f_{2} = x^{3}\right\}$$$.

Wronskian diberikan oleh determinan berikut: $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}f_{1}\left(x\right) & f_{2}\left(x\right)\\f_{1}^{\prime}\left(x\right) & f_{2}^{\prime}\left(x\right)\end{array}\right|.$$$

Dalam kasus ini, $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}x & x^{3}\\\left(x\right)^{\prime } & \left(x^{3}\right)^{\prime }\end{array}\right|.$$$

Temukan turunan (untuk langkah-langkahnya, lihat kalkulator turunan): $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}x & x^{3}\\1 & 3 x^{2}\end{array}\right|$$$

Temukan determinan (untuk langkah-langkah, lihat kalkulator determinan): $$$\left|\begin{array}{cc}x & x^{3}\\1 & 3 x^{2}\end{array}\right| = 2 x^{3}$$$

Wronskian sama dengan $$$2 x^{3}$$$A.