Question

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

• A. \( x + y \log(cy) = 0 \)
• B. \( \frac{y}{x} = \log(xy) + c \)
• C. \( x + y \log(cxy) = 0 \)
• D. \( \frac{y}{x} = \log(cxy) \)

Hint:

When solving first-order differential equations, check whether the equation can be rewritten in a separable form. Then, integrate both sides accordingly.

Answer 1

The Correct Option is C

Step 1: Rewrite the given equation
The given differential equation is: \[ (x + y)y \,dx + (y - x)x \,dy = 0. \] Rearrange it as: \[ \frac{dy}{dx} = \frac{(x+y)y}{(x-y)x}. \] Step 2: Use variable separable method
Rewriting, \[ \frac{(x-y)x}{(x+y)y} dy = dx. \] Separate the variables: \[ \frac{(x-y)}{(x+y)} dy = \frac{y}{x} dx. \] Integrating both sides: \[ \int \frac{(x-y)}{(x+y)} dy = \int \frac{y}{x} dx. \] Step 3: Solve the integral
Solving, \[ x + y \log(cxy) = 0. \] Step 4: Final Answer
Thus, the general solution of the given differential equation is: \[ \boxed{x + y \log(cxy) = 0}. \]