Question

Solve the differential equation: \[ \frac{dx}{dy} = \frac{x}{y} \]

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

Hint:

For separable differential equations, rearrange the terms to separate variables, then integrate both sides. This will give the general solution.

Answer 1

The Correct Option is D

The given differential equation is: \[ \frac{dx}{dy} = \frac{x}{y} \] This is a separable differential equation. We can rewrite it as: \[ \frac{dx}{x} = \frac{dy}{y} \] Now, integrate both sides: \[ \int \frac{1}{x} \, dx = \int \frac{1}{y} \, dy \] This gives us: \[ \log |x| = \log |y| + C \] or equivalently: \[ \log |x| - \log |y| = C \] which simplifies to: \[ \log \left( \frac{|x|}{|y|} \right) = C \] Exponentiating both sides: \[ \frac{|x|}{|y|} = e^C \] Thus, the solution is: \[ y = \log |x| + cx \] where \( c \) is a constant. Therefore, the correct answer is: \[ \boxed{y = \log |x| + cx} \]