1. ## Wronskian computation

Compute the Wronskian of the given set of functions:

cos ax, sin ax, x, a does not =0, in any interval.

Could someone help me with this?

Thanks,

Newt

2. Originally Posted by Jim Newt
Compute the Wronskian of the given set of functions:

cos ax, sin ax, x, a does not =0, in any interval.

Could someone help me with this?

Thanks,

Newt

$\displaystyle \begin{vmatrix} \cos(ax) && \sin(ax) && x \\ -a \sin(ax) && a \cos(ax) && 1 \\ -a^2 \cos(ax) &&-a^2 \sin(ax) && 0 \\ \end{vmatrix}$

Now factor an $\displaystyle a^2$ out of row 3

$\displaystyle a^2 \begin{vmatrix} \cos(ax) && \sin(ax) && x \\ -a \sin(ax) && a \cos(ax) && 1 \\ - \cos(ax) &&- \sin(ax) && 0 \\ \end{vmatrix}$

add row3 to row 1 and expand the det along the 1st row

$\displaystyle a^2 \begin{vmatrix} 0 && 0 && x \\ -a \sin(ax) && a \cos(ax) && 1 \\ - \cos(ax) &&- \sin(ax) && 0 \\ \end{vmatrix}= a^2 \cdot x \begin{vmatrix} -a \sin(ax) && a \cos(ax) \\ -\cos(ax) && - \sin(ax) \\ \end{vmatrix}$

$\displaystyle =a^2 \cdot x (a \sin^{2}(ax)+a \cos^{2}(ax))=a^2 \cdot x \cdot (a)=a^3 \cdot x$

so the wronskian is not ALWAYS zero so the set is linearly independant.