Question

Use operations and cofactor expansion to evaluate the Wronskian of the set of functions: {e^x , xe^x , x^2 e^x}

Answer 1

\[W=W(e^x,xe^x,x^2e^x)=\begin{vmatrix}e^x&xe^x&x^2e^x\\
e^x&(x+1)e^x&(x^2+2x)e^x\\
e^x&(x+2)e^x&(x^2+4x+2)e^x\end{vmatrix}\]
Recall that \(\det(kA)=k\det A\), so
\[W=e^x\begin{vmatrix}1&x&x^2\\
1&x+1&x^2+2x\\
1&x+2&x^2+4x+2\end{vmatrix}\]
You have 6 choices for the cofactor expansion. At random, I'll demonstrate the expansion along the last column:
\[W=e^x\left(x^2\begin{vmatrix}1&x+1\\1&x+2\end{vmatrix}-(x^2+2x)\begin{vmatrix}1&x\\1&x+2\end{vmatrix}+(x^2+4x+2)\begin{vmatrix}1&x\\1&x+1\end{vmatrix}\right)\]
From here it's a simple matter of computing determinants of \(2\times2\) matrices, which you'll recall is given by
\[\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc\]