Write short notes on the following: Reimer-Tiemann's reaction
Step 1: Reimer-Tiemann reaction is used to introduce a formyl (-CHO) group into phenol.
Step 2: Reaction: \[ C_6H_5OH + CHCl_3 + NaOH \rightarrow o-\text{hydroxybenzaldehyde} + p-\text{hydroxybenzaldehyde} \]
Step 3: The reaction mechanism involves the formation of a dichlorocarbene intermediate, which undergoes electrophilic substitution at the ortho- and para-positions of phenol.
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 $