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Suppose you solved a second-order equation by rewriting it as a system and found two scalar solutions: $y = e^{3 x}$ and $z = e^{4 x}$ . Think of the corresponding vector solutions $\displaystyle \boldsymbol{ \vec{y}_1 }$ and $\displaystyle \boldsymbol{ \vec{y}_2 }$ and use the Wronskian to show that the solutions are linearly independent.

Wronskian = $\mathrm{det}$

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=

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