Question

The general solution of the differential equation \( x \, dy + y \, dx = 0 \) is:

• A. \( xy = c \)
• B. \( x + y = c \)
• C. \( x^2 + y^2 = c^2 \)
• D. \( \log y = \log x + c \)

Hint:

For separable differential equations, rearrange terms and integrate both sides.

Answer 1

The Correct Option is A

Step 1: Rewrite the equation
The given equation \( x \, dy + y \, dx = 0 \) can be written as: \[ \frac{dy}{y} + \frac{dx}{x} = 0. \]
Step 2: Integrate both sides
\[ \int \frac{dy}{y} + \int \frac{dx}{x} = 0 \implies \ln|y| + \ln|x| = C. \]
Step 3: Simplify the solution
Combine logarithms: \[ \ln|xy| = C \implies xy = e^C = c. \]
Step 4: Verify the options
The solution is \( xy = c \), which matches option (A).