Question

The solution of the differential equation \( \frac{dy}{dx} = \frac{1}{\log y} \) is:

• A. \( \log y = x + c \)
• B. \( y \log y - y = x + c \)
• C. \( \log y - y = x + c \)
• D. \( y \log y + y = x + c \)

Hint:

For differential equations involving \( \frac{dy}{dx} \), rewrite to separate variables, integrate, and simplify.

Answer 1

The Correct Option is B

Step 1: Rewrite the given differential equation. The given equation is: \[ \frac{dy}{dx} = \frac{1}{\log y} \] Rewriting, we obtain: \[ \log y \, dy = dx \] Step 2: Integrate both sides. \[ \int \log y \, dy = \int dx \] Using integration by parts for \( \int \log y \, dy \), let \( u = \log y \) and \( dv = dy \), so: \[ \int \log y \, dy = y \log y - \int y \, dy = y \log y - y + C_1 \] Step 3: Substituting the integration result. \[ y \log y - y = x + C_1 \] Letting \( C = C_1 \), the final solution becomes: \[ y \log y - y = x + C \] Final Answer: \[ \boxed{y \log y - y = x + c} \]