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:

To identify the type of differential equation, rewrite it in standard forms and compare the coefficients.

Answer 1

The Correct Option is C

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.