Given DE: \(x\frac{dy}{dx}\) = y + \(x\, tan \frac{y}{x}\) Now, Dividing both sides by x, we get, \(\frac{dy}{dx}\) = \(\frac{y}{x}\) + tan \(\frac{y}{x}\)
Let y = vx Differentiating both sides, dy/dx = v + xdv/dx Now, our differential equation becomes, v + \(x \frac{dv}{dx}\) = v + tan v
On separating the variables, we get, \(\frac{dv}{tan\, v}\) = \(\frac{dx}{x}\) Integrating both sides, we get, sin v = Cx
Putting the value of v we get, sin \((\frac{y}{x})\) = Cx .