Question

In Problems 8 to 15,use to show that the given functions are linearly independent.

15.${\mathbf{e}}^x,{\mathbf{e}}^{ix},\cosh x$

Short answer

It has been shown that the functions$e^x,e^{ix},\cosh x$ are linearly independent.

Step-by-step solution

Definition of linearly independent functions

The functions$f_1\left(x\right),f_2\left(x\right),\dots ,f_n\left(x\right)$are linearly independent if the determinant

$N=\left|\begin{array}{cccc}{f}_1\left(x\right)& f_2\left(x\right)& \cdots & f_n\left(x\right)\\ {f_1}^{\mathrm{\prime }}\left(x\right)& {f_2}^{\mathrm{\prime }}\left(x\right)& \cdots & f_n^{\mathrm{\prime }}\left(x\right)\\ ⋮& ⋮& \ddots & ⋮\\ f_1^{(n-1)}\left(x\right)& f_2^{(n-1)}\left(x\right)& \cdots & f_n^{(n-1)}\left(x\right)\end{array}\right|\ne 0$

Here, W is called the Wronksian of functions.

Use the Wronksian to show that the given functions are linearly independent. 

Find the derivatives of the function of order $f_1\left(x\right)=e^x\text{of order}n-1\text{.}$

$\begin{array}{r}{f}_1\left(x\right)=e^x\\ f_1^{\mathrm{\prime }}\left(x\right)=e^x\\ f_1^{\prime \prime }\left(x\right)=e^x\end{array}$

Find the derivatives of the function of order$f_2\left(x\right)=e^{ix}\text{of order}n-1$

$\begin{array}{r}{f}_2\left(x\right)=e^{ix}\\ f_2^{\prime \prime }\left(x\right)=i{e}^{ix}\\ f_2^{\ast }\left(x\right)=-e^{ix}\end{array}$

Find the derivatives of the function of order$f_3\left(x\right)=\cosh x\text{of order}n-1$

$\begin{array}{r}{f}_3\left(x\right)=\cosh x\\ f_3^{\mathrm{\prime }}\left(x\right)=\sinh x\\ f_3^{\mathrm{\prime }}\left(x\right)=\cosh x\end{array}$