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Found in: Page 326

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.$\left\{{e}^{3x}, {e}^{5x}, {e}^{-x}\right\}$ on $\left(-\infty , \infty \right)$

Thus, $\left\{{\mathbf{e}}^{\mathbf{3}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{5}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{-}\mathbf{x}}\right\}$ are linearly independent on $\left(\mathbf{-}\infty \mathbf{,} \infty \right)$.

See the step by step solution

## Step 1:Using the concept of Wronskian

The given function is $\left\{{\mathbf{e}}^{\mathbf{3}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{5}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{-}\mathbf{x}}\right\}.$

Apply the concept of Wronskian,

$W\left[{f}_{1},{f}_{2},\dots ,{f}_{n}\right]=\left|\begin{array}{cccc}{f}_{1}\left(x\right)& {f}_{2}\left(x\right)& \dots & {f}_{n}\left(x\right)\\ {f}_{1}\text{'}\left(x\right)& {f}_{2}\text{'}\left(x\right)& \dots & {f}_{n}\text{'}\left(x\right)\\ ⋮& ⋮& & ⋮\\ {{f}_{1}}^{n-1}\left(x\right)& {{f}_{2}}^{n-1}\left(x\right)& \cdots & {{f}_{n}}^{n-1}\left(x\right)\end{array}\right|$

Therefore,

$W\left[{e}^{3x},{e}^{5x},{e}^{-x}\right]=\left|\begin{array}{ccc}{e}^{3x}& {e}^{5x}& {e}^{-x}\\ 3{e}^{3x}& 5{e}^{5x}& -{e}^{-x}\\ 9{e}^{3x}& 25{e}^{5x}& {e}^{-x}\end{array}\right|$

Solve the above equation,

$W\left[{e}^{3x},{e}^{5x},{e}^{-x}\right]={e}^{3x}×{e}^{5x}×{e}^{-x}\left|\begin{array}{ccc}1& 1& 1\\ 3& 5& -1\\ 9& 25& 1\end{array}\right|\phantom{\rule{0ex}{0ex}}={e}^{7x}\left[1\left(30\right)-1\left(12\right)+1\left(30\right)\right]\phantom{\rule{0ex}{0ex}}={e}^{7x}\left[\left[48\right]\right]\phantom{\rule{0ex}{0ex}}=48{e}^{7x}$

## Step 2:Check the linearly independent or dependent

Since the above result is $\mathbf{48}{\mathbf{e}}^{\mathbf{7}\mathbf{x}}\ne \mathbf{0}$ $\left(\forall \mathbf{x}\right)$.

Therefore, $\left\{{\mathbf{e}}^{\mathbf{3}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{5}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{-}\mathbf{x}}\right\}$ are linearly independent on $\left(\mathbf{-}\infty \mathbf{,} \infty \right)$.