Wronskin determinantti funktioille $$$x$$$, $$$x^{3}$$$

Laske Wronskin determinantti funktioille $$$\left\{f_{1} = x, f_{2} = x^{3}\right\}$$$.

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

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

Laske derivaatat (vaiheet: katso derivointilaskuri): $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}x & x^{3}\\1 & 3 x^{2}\end{array}\right|$$$.

Laske determinantti (vaiheet: ks. determinanttilaskin): $$$\left|\begin{array}{cc}x & x^{3}\\1 & 3 x^{2}\end{array}\right| = 2 x^{3}$$$

Wronskin determinantti on $$$2 x^{3}$$$A.