Step 1: Potassium permanganate (\( KMnO_4 \)) is prepared from pyrolusite (\( MnO_2 \)) via the following steps:
Step 2: Oxidation of Pyrolusite: \[ 2 MnO_2 + 4 KOH + O_2 \rightarrow 2 K_2MnO_4 + 2 H_2O \]
Step 3: Disproportionation of Manganate: \[ 3 K_2MnO_4 + 2 H_2O \rightarrow 2 KMnO_4 + MnO_2 + 4 KOH \]
Step 4: The structural formula of the permanganate ion (\( MnO_4^- \)): \[ \text{A tetrahedral structure with Mn in +7 oxidation state.} \]
Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where
\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]
(b) Order of the differential equation: $ 5x^3 \frac{d^3y}{dx^3} - 3\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0 $