Question

\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:

• A. variable separable differential equation
• B. homogeneous differential equation
• C. first-order linear differential equation
• D. differential equation whose degree is not defined

Hint:

First-order linear differential equations take the form \( \frac{dy}{dx} + P(x)y = Q(x) \).

Answer 1

The Correct Option is C

The equation is rewritten as: \[x \log x \frac{dy}{dx} + y = 2 \log x \quad \Rightarrow \quad \frac{dy}{dx} + \frac{y}{x \log x} = \frac{2}{x \log x}.\] This is a first-order linear differential equation, which can be expressed in the form: \[\frac{dy}{dx} + P(x)y = Q(x).\]
Final Answer: \( \boxed{ {first-order linear differential equation}} \)