Step 1: Rewrite the equation
The given equation is: \[ x \log x \frac{dy}{dx} + y = 2 \log x. \] Rearranging: \[ \frac{dy}{dx} + \frac{y}{x \log x} = \frac{2}{x \log x}. \]
Step 2: Check the form of the equation
This is a first-order linear differential equation of the form: \[ \frac{dy}{dx} + P(x)y = Q(x), \] where \( P(x) = \frac{1}{x \log x} \) and \( Q(x) = \frac{2}{x \log x} \).
Step 3: Conclude the result
The equation is a first-order linear differential equation.
A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum.
On the basis of the above information, answer the following questions :
Find \( \frac{dS}{dx} \).
Find the interval in which $f(x) = x + \frac{1}{x}$ is always increasing, $x \neq 0$.