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

Wronskian = $\mathrm{det}$

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=

The test for linear independence of the set $\lbrace e^{x+3}, e^{x+4} \rbrace$ using the Wronskian is inconclusive because the Wronskian is for all $x$ .

If the functions $e^{x+3}$ and $e^{x+4}$ are linearly dependent, find a nontrivial solution to the equation below. If they are linearly independent, enter all zeros to indicate that the only solution to the equation is the trivial solution.

$(e^{x+3}) +$ $(e^{x+4}) = 0.$