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Math Help - Wronskian computation

  1. #1
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    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
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Jim Newt View Post
    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

    \begin{vmatrix}<br />
\cos(ax) && \sin(ax) && x \\<br />
-a \sin(ax) && a \cos(ax) && 1 \\<br />
-a^2 \cos(ax) &&-a^2 \sin(ax) && 0 \\<br />
\end{vmatrix}

    Now factor an a^2 out of row 3

    a^2 \begin{vmatrix}<br />
\cos(ax) && \sin(ax) && x \\<br />
-a \sin(ax) && a \cos(ax) && 1 \\<br />
- \cos(ax) &&- \sin(ax) && 0 \\<br />
\end{vmatrix}

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

    a^2 \begin{vmatrix}<br />
0 && 0 && x \\<br />
-a \sin(ax) && a \cos(ax) && 1 \\<br />
- \cos(ax) &&- \sin(ax) && 0 \\<br />
\end{vmatrix}= a^2 \cdot x \begin{vmatrix}<br />
-a \sin(ax) && a \cos(ax) \\<br />
-\cos(ax) && - \sin(ax) \\<br />
\end{vmatrix}

    =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.
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