Q14E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

$\left\{x, {\mathrm{xe}}^x, 1\right\}$ on $\left(-\infty , \infty \right)$

Step-by-Step Solution

Answer

Thus, the function  $\left\{\mathbf{x}\mathbf{,} {\mathbf{xe}}^{\mathbf{x}}\mathbf{,} \mathbf{1}\right\}$ is linearly independent on $\left(\mathbf{-}\infty \mathbf{,} \infty \right)$.

Step 1:Using the concept of Wronskian

The given function is $\left\{\mathbf{x}\mathbf{,} {\mathbf{xe}}^{\mathbf{x}}\mathbf{,} \mathbf{1}\right\}.$

 

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,

 

$W\left[x,x{e}^x,1\right]=\left|\begin{array}{ccc}x& x{e}^x& 1\\ 1& x{e}^x+e^x& 0\\ 0& x{e}^x+2e^x& 0\end{array}\right|$

 

Solve the above equation,

 $$\begin{gathered} \mathbf{W}\left[\mathbf{x}\mathbf{,} {\mathbf{xe}}^{\mathbf{x}}\mathbf{,} \mathbf{1}\right]\mathbf{=}\mathbf{x}\left(\mathbf{0}\right)\mathbf{-}{\mathbf{xe}}^{\mathbf{x}}\left(\mathbf{0}\right)\mathbf{+}\mathbf{1}\left({\mathbf{xe}}^{\mathbf{x}}\mathbf{+}\mathbf{2}{\mathbf{e}}^{\mathbf{x}}\right) \\ \mathbf{=}{\mathbf{xe}}^{\mathbf{x}}\mathbf{+}\mathbf{2}{\mathbf{e}}^{\mathbf{x}} \end{gathered}$$

Step 2: Check the linearly independent or dependent

The above function is not equal to zero $\forall x$..

 

Thus, the function  $\left\{\mathbf{x}\mathbf{,} {\mathbf{xe}}^{\mathbf{x}}\mathbf{,} \mathbf{1}\right\}$ are linearly independent on $\left(\mathbf{-}\infty \mathbf{,} \infty \right)$.