When analyzing first-order linear differential equations, carefully identify the degree and whether the equation is homogeneous. For solving, try rearranging the equation and use the method of integrating factors or other appropriate techniques to find the solution. It is essential to verify the solution by substituting it back into the original equation. In this case, simplifying and integrating the equation gives us the correct solution.
The differential equation provided is:
\((x \log_e x) \, dy = (\log_e x - y) \, dx.\)
Upon rearrangement:
\((x \log_e x) \, dy + y \, dx = \log_e x \, dx.\)
This equation is a first-order linear differential equation.
(A) The degree of the given differential equation is 1. This is accurate, as the highest power of the derivatives is 1.
(B) The equation is not homogeneous. This is incorrect because it fails to meet the criteria for a homogeneous equation. A homogeneous differential equation requires both sides to be expressible as a function of \( \frac{dy}{dx} \), a form which this equation does not adopt.
(C) The solution is \( 2y \log_e x + A = (\log_e x)^2 \), where \( A \) is an arbitrary constant. This is the correct solution. Integrating and simplifying yields the solution:
\(\int \frac{dy}{dx} = \log_e x \implies 2y \log_e x = (\log_e x)^2 + A.\)
Therefore, (C) is correct.
(D) The solution is \( 2y \log_e x + A = \log_e (\log_e x) \), where \( A \) is an arbitrary constant. This is incorrect, as the solution obtained from the differential equation does not match this form.
Consequently, the correct answer is: (A) - (C) only.