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Use the Wronskian to show that the functions $y_1 = e^{3 x}$ and $y_2 = e^{4 x}$ are linearly independent.

Wronskian = $\mathrm{det}$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$			$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

=

These functions are linearly independent because the Wronskian is for all $x$ .