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Odd Function

Functions Px,...

Question

Functions $P (x), Q (x), R (x)$ are differentiable on some open interval around $0$ and satisfy the below equations as well as the initial conditions.
$P^{'} (x) = 2 P^{2} (x) Q (x) R (x) + \frac{1}{Q (x) R (x)},$ $P (0) = 1$
$Q^{'} (x) = P (x) Q^{2} (x) R (x) + \frac{4}{P (x) R (x)},$ $Q (0) = 1$
$R^{'} (x) = 3 P (x) Q (x) R^{2} (x) + \frac{1}{P (x) Q (x)},$ $R (0) = 1.$
Then $P (x) Q (x) R (x) = tan (n x + \frac{π}{4}) .$ The value of $n$ is

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Solution

$P^{'} (x) = 2 P^{2} (x) \cdot Q (x) \cdot R (x) + \frac{1}{Q (x) \cdot R (x)} \dots (1)$
$Q^{'} (x) = P (x) \cdot Q^{2} (x) \cdot R (x) + \frac{4}{P (x) \cdot R (x)} \dots (2)$
$R^{'} (x) = 3 P (x) \cdot Q (x) \cdot R^{2} (x) + \frac{1}{P (x) \cdot Q (x)} \dots (3)$

Multiply $Q (x) R (x)$ in $(1),$ $P (x) R (x)$ in $(2)$ and $P (x) Q (x)$ in $(3)$
$P^{'} (x) Q (x) R (x) = 2 P^{2} (x) \cdot Q^{2} (x) \cdot R^{2} (x) + 1 \dots (4) P (x) Q^{'} (x) R (x) = P^{2} (x) \cdot Q^{2} (x) \cdot R^{2} (x) + 4 \dots (5) P (x) Q (x) R^{'} (x) = 3 P^{2} (x) \cdot Q^{2} (x) \cdot R^{2} (x) + 1 \dots (6)$

Now, $(4) + (5) + (6)$
${[P (x) Q (x) R (x)]}^{'} = 6 {[P (x) Q (x) R (x)]}^{2} + 6$
Let $P (x) Q (x) R (x) = t$
$\Rightarrow {[P (x) Q (x) R (x)]}^{'} d x = d t$
$\Rightarrow \frac{d t}{t^{2} + 1} = 6 d x \Rightarrow {tan}^{- 1} t = 6 x + C x = 0 \Rightarrow C = \frac{π}{4}$
$∴ P (x) Q (x) R (x) = tan (6 x + \frac{π}{4})$

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