\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:
Show Hint
- variable separable differential equation
- homogeneous differential equation
- first order linear differential equation
- differential equation whose degree is not defined
The Correct Option is C
Solution and Explanation
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.
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