1. The problem states that $$\int \frac{2}{y} \, dy = \int \frac{1}{x} \, dx$$ and asks to find the expression for $$\ln y^2$$ in terms of $$x$$ plus a constant $$c$$. 2. Compute the integral on the left side: $$\int \frac{2}{y} \, dy = 2 \int \frac{1}{y} \, dy = 2 \ln |y| + C_1$$ 3. Compute the integral on the right side: $$\int \frac{1}{x} \, dx = \ln |x| + C_2$$ 4. Equate the two integrals (absorbing constants into a single constant $$c$$): $$2 \ln |y| = \ln |x| + c$$ 5. Use logarithm properties to rewrite the left side: $$2 \ln |y| = \ln |y|^2$$ 6. Therefore, the equation becomes: $$\ln |y|^2 = \ln |x| + c$$ 7. This matches option (a) $$\ln |x|$$ plus a constant. Final answer: $$\ln y^2 = \ln |x| + c$$