If f and g are two real valued functions differentiable over R such that f '(x) = g(x) + \frac{x}{f \left(\right. x \left.\right)} and g'(x) + f (x) = \frac{x}{g \left(\right. x \left.\right)}. If f (1) = 1 and g(1) = \sqrt{2}, then find the value of f 2(2) + g2(2).

Engineering

Mathematics

Basic Rule of Differentiation

Question

If f and g are two real valued functions differentiable over R such that f '(x) = g(x) + $\frac{x}{f (x)}$ and g'(x) + f (x) = $\frac{x}{g (x)}$ . If f (1) = 1 and g(1) = $\sqrt{2}$ , then find the value of f ²(2) + g²(2).

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Let h(x) = f ²(x) + g²(x)
h'(x) = 2f (x) · f '(x) + 2 g(x) · g'(x)
   = 2f (x) $(g (x) + \frac{x}{f (x)}) + 2 g (x) (- f (x) + \frac{x}{g (x)}) = 4 x$
⇒    h(x) = 2x² + C
       h(1) = 2 + c = 3 ⇒ c = 1
$∴$    h(2) = 9