Question

If $y=\sin px$ prove that
$\Delta =\left|\begin{array}{ccc}y& y_1& y_2\\ y_3& y_4& y_5\\ y_6& y_7& y_8\end{array}\right|=0.$
where $y_r$, means $r-th$ differential coefficient of $y$.

Answer 1

$\Delta =\left|\begin{array}{ccc}\sin px& \cos px& -p^2\sin px\\ -p^3\cos px& p^3\sin px& p^5\cos px\\ -p^6\sin px& {-p}^7\cos px& p^8\sin px\end{array}\right|$
$={-p}^3\left|\begin{array}{ccc}\sin px& \cos px& \sin px\\ -p^3\cos px& p^3\sin px& -p^3\cos px\\ -p^6\sin px& {-p}^6\cos px& -p^6\sin px\end{array}\right|$
$=0$, since $C_1$, $C_3$ are identical.