If $$y_1(x)=\sin 2x$$ and $$y_2(x)=\cos 2x$$ are two solutions of $$y^{,,}+4y=0,$$ show that $y_1(x)$ and $y_2(x)$ are linearly independent solutions.
I think to prove linearly independent,the equation will be
$$c_1\sin 2x+c_2\cos 2x=0$$,where $c_1$ and $c_2$ have to be equal to zero,but how to prove that??,please help me..
Best Answer
One way to show the independence of solutions is to show that the Wronskian is not identically zero.
$$ W= \det \begin {bmatrix}y_1&y_2\\y'_1&y'_2\end{bmatrix}\not = 0$$
You can take it from here.