Question

In Problems 15–22 Determine whether the given set of functions is linearly independent on the interval $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$ .

${\mathbf{f}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{x}}\mathbf{,}\mathbf{}{\mathbf{f}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{,}\mathbf{}{\mathbf{f}}_{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{sin}\mathbf{}\mathbf{hx}$.

Short answer

The solution of set of functions on the given interval $(-\infty ,\infty )$is linearly dependent.

Step-by-step solution

Deriving the given solution by wronskian equation

The given solution is,

${\mathrm{f}}_1\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}},{\mathrm{f}}_2\left(\mathrm{x}\right)={\mathrm{e}}^{-\mathrm{x}}\mathrm{and}{\mathrm{f}}_3\left(\mathrm{x}\right)=\mathrm{sinhx}$

We have to determine if this set is linearly independent on the interval$(-\infty ,\infty )$ or not as the following technique:

First, we have to obtain the wronskian of this set of functions as

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& \left|\begin{array}{ccc}{\mathrm{f}}_1& {\mathrm{f}}_2& {\mathrm{f}}_3\\ {\mathrm{f}\text{'}}_1& {\mathrm{f}\text{'}}_2& {\mathrm{f}\text{'}}_3\\ {\mathrm{f}}_1\text{'}\text{'}& {\mathrm{f}}_2\text{'}\text{'}& {\mathrm{f}}_3\text{'}\text{'}\end{array}\right|\\ & =& \left|\begin{array}{ccc}{\mathrm{e}}^{\mathrm{x}}& {\mathrm{e}}^{-\mathrm{x}}& \sin \mathrm{hx}\\ {\mathrm{e}}^{\mathrm{x}}& -{\mathrm{e}}^{-\mathrm{x}}& \cos \mathrm{hx}\\ {\mathrm{e}}^{\mathrm{x}}& {\mathrm{e}}^{-\mathrm{x}}& \sin \mathrm{hx}\end{array}\right|\end{array}$

Second, we have to find the determinate of this matrix as

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{e}}^{\mathrm{x}},{\mathrm{e}}^{-\mathrm{x}},\sin \mathrm{hx}\right)& =& {\mathrm{e}}^{\mathrm{x}}\times \left|\begin{array}{cc}-{\mathrm{e}}^{-\mathrm{x}}& \cos \mathrm{hx}\\ {\mathrm{e}}^{-\mathrm{x}}& \sin \mathrm{hx}\end{array}\right|-{\mathrm{e}}^{-\mathrm{x}}\times \left|\begin{array}{cc}{\mathrm{e}}^{\mathrm{x}}& \cos \mathrm{hx}\\ {\mathrm{e}}^{\mathrm{x}}& \sin \mathrm{hx}\end{array}\right|+\sin \mathrm{hx}\times \left|\begin{array}{cc}{\mathrm{e}}^{\mathrm{x}}& -{\mathrm{e}}^{-\mathrm{x}}\\ {\mathrm{e}}^{\mathrm{x}}& {\mathrm{e}}^{-\mathrm{x}}\end{array}\right|\\ & =& {\mathrm{e}}^{\mathrm{x}}\left[-{\mathrm{e}}^{-\mathrm{x}}\sin \mathrm{hx}-{\mathrm{e}}^{-\mathrm{x}}\cos \mathrm{hx}\right]-{\mathrm{e}}^{-\mathrm{x}}\left[{\mathrm{e}}^{\mathrm{x}}\sin \mathrm{hx}-{\mathrm{e}}^{\mathrm{x}}\cos \mathrm{hx}\right]+\sin \mathrm{hx}[1-(-1\left)\right]\\ & =& (-\sin \mathrm{hx}-\cos \mathrm{hx})-(\sin \mathrm{hx}-\cos \mathrm{hx})+2\sin \mathrm{hx}\\ & =& (2\sin \mathrm{hx}-2\sin \mathrm{hx})+(\cos \mathrm{hx}-\cos \mathrm{hx})\\ & =& 0\end{array}$

Since the determinate of the wronskian of the given set of functions equal Zero,

Then this set of function is linearly dependent.