Chapter 9: First-Order Differential Equations - Section 9.2 - First-Order Linear Equations - Exercises 9.2 - Page 537: 23

(a) is incorrect and (b) is correct.

Here, we have $x \int \ln |x| dx=x(\ln |x|)+C$ This can be re-arranged as: $x \int \ln |x| dx=x\ln |x|+Cx$ Therefore, we conclude that the solution $(a)$ is incorrect because the arbitrary constant depend on the values of $C$. whereas, the solution $(b)$ is correct because it is same as the solution.