The general solution of the differential equation \(x\left(\displaystyle\frac{dy}{dx}\right) = y + x\tan\left(\displaystyle\frac{y}{x}\right)\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\sin\left(\displaystyle\frac{y}{x}\right) = cx\), where c is an arbitrary constant
Dividing by x:
\( \displaystyle \frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right) \)
The right side depends only on \(\frac{y}{x}\), so this is a homogeneous differential equation. Let \(y = vx\):
\( \displaystyle \frac{dy}{dx} = v + x\frac{dv}{dx} \)
\( \displaystyle v + x\frac{dv}{dx} = v + \tan v \)
Separating the variables:
\( \displaystyle \cot v\,dv = \frac{dx}{x} \)
Integrating, with the constant written as \(\log|c|\):
\(\log|\sin v| = \log|x| + \log|c|\)
\(\sin v = cx\)
Putting back \(v = \frac{y}{x}\):
\( \displaystyle \sin\left(\frac{y}{x}\right) = cx \)