Wronskian of $$$x$$$, $$$x^{3}$$$
Your Input
Calculate the Wronskian of $$$\left\{f_{1} = x, f_{2} = x^{3}\right\}$$$.
Solution
The Wronskian is given by the following determinant: $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}f_{1}\left(x\right) & f_{2}\left(x\right)\\f_{1}^{\prime}\left(x\right) & f_{2}^{\prime}\left(x\right)\end{array}\right|.$$$
In our case, $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}x & x^{3}\\\left(x\right)^{\prime } & \left(x^{3}\right)^{\prime }\end{array}\right|.$$$
Find the derivatives (for steps, see derivative calculator): $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}x & x^{3}\\1 & 3 x^{2}\end{array}\right|$$$.
Find the determinant (for steps, see determinant calculator): $$$\left|\begin{array}{cc}x & x^{3}\\1 & 3 x^{2}\end{array}\right| = 2 x^{3}$$$.
Answer
The Wronskian equals $$$2 x^{3}$$$A.