Question

In Problems$8\text{to}15,\text{use}\left(8.5\right)$to show that the given functions are linearly independent.

$\text{13.}\sin x,\sin 2x$

Short answer

It has been shown that the functions $\sin x,\sin 2x$are linearly independent except at

$x=\left(n+\frac{1}{2}\right)\mathrm{\Pi }$

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

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

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 $f_1\left(x\right)=\sin x\text{of order}1$

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

Find the derivatives of the function of order $f_2\left(x\right)=\sin 2x\text{of order}1$

$\begin{array}{r}{f}_2\left(x\right)=\sin 2x\\ f_2^{\mathrm{\prime }}\left(x\right)=2\cos 2x\end{array}$

Substitute the derivatives in the Wronksian formula and simplify as follows:

localid="1657426787997" $\begin{array}{r}W=\left|\begin{array}{cc}\sin x& \sin 2x\\ \cos x& 2\cos 2x\end{array}\right|\\ =2\sin x\cos 2x-\sin 2x\cos x\end{array}$

Use the trigonometric identities $\cos 2x={\cos }^{2}x-{\sin }^{2}x\text{and}\sin 2x=2\sin x\cos x$