Question

Find the Wronskian of the following set of function
{sin 3x,cos 3x}

Answer 1

We consider $y_1=\sin 3x$ and $y_2=\cos 3x$ . Then the requested wronskian is:

$$W(y_1,y_2)=\begin{vmatrix} y_1 & y_2\\ y_1' & y_2' \end{vmatrix}=y_1y_2'-y_1'y_2$$

$$\left. \begin{array}{l} y_1=\sin 3x\Rightarrow y_1'=3\cos 3x\\ y_2=\cos 3x\Rightarrow y_2'=-3\sin 3x \end{array} \right\}$$

Substituting the latter into the Wronskian, we obtain:

$$W(\sin 3x, \cos 3x)=\begin{vmatrix} \sin 3x & \cos 3x\\ 3\cos 3x & -3\sin 3x \end{vmatrix}=-3\sin^23x+3\cos^2 3x=$$

$$=3(\cos^2 3x-\sin^2 3x)=3\cos (2\cdot 3x)=\boxed{3\cos 6x}$$