The general solution of the differential equation \((x + y)\,dy = y\,dx\), \(y \ne 0\), is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(x = y\log |y| + Cy\)
Writing the equation with x as the dependent variable:
\( \displaystyle \frac{dx}{dy} = \frac{x + y}{y} = \frac{x}{y} + 1 \)
The right side is a function of \(\frac{x}{y}\), so this is a homogeneous equation of the form \(\frac{dx}{dy} = h\left(\frac{x}{y}\right)\), solved by the substitution \(x = vy\):
\( \displaystyle \frac{dx}{dy} = v + y\frac{dv}{dy} \)
\( \displaystyle v + y\frac{dv}{dy} = v + 1 \)
\( \displaystyle dv = \frac{dy}{y} \)
\( v = \log |y| + C \)
Putting \(v = \frac{x}{y}\) back:
\( x = y\log |y| + Cy \)