Wronskin determinantti funktioille $$$x$$$, $$$\frac{1}{x^{5}}$$$

Laske Wronskin determinantti funktioille $$$\left\{f_{1} = x, f_{2} = \frac{1}{x^{5}}\right\}$$$.

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 $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}x & \frac{1}{x^{5}}\\\left(x\right)^{\prime } & \left(\frac{1}{x^{5}}\right)^{\prime }\end{array}\right|.$$$

Laske derivaatat (vaiheet: katso derivointilaskuri): $$$W{\left(f_{1},f_{2} \right)}\left(x\right) = \left|\begin{array}{cc}x & \frac{1}{x^{5}}\\1 & - \frac{5}{x^{6}}\end{array}\right|$$$.

Laske determinantti (vaiheet: ks. determinanttilaskin): $$$\left|\begin{array}{cc}x & \frac{1}{x^{5}}\\1 & - \frac{5}{x^{6}}\end{array}\right| = - \frac{6}{x^{5}}$$$

Wronskin determinantti on $$$- \frac{6}{x^{5}}$$$A.