If frac{dy}{dx} = frac{y + x tan(frac{y}{x})} | vedclass

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(B) Given the homogeneous differential equation:$\frac{dy}{dx} = \frac{y + x 	an(\frac{y}{x})}{x} \quad \dots(i)$Let $y = vx$, then $\frac{dy}{dx} =

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$cx$

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(B) Given the homogeneous differential equation:$\frac{dy}{dx} = \frac{y + x 	an(\frac{y}{x})}{x} \quad \dots(i)$Let $y = vx$, then $\frac{dy}{dx} = v + x \frac{dv}{dx}$.Substituting these into equation $(i)$:$v + x \frac{dv}{dx} = \frac{vx + x 	an v}{x}$$v + x \frac{dv}{dx} = v + 	an v$$x \frac{dv}{dx} = 	an v$Separating the variables:$\cot v \, dv = \frac{dx}{x}$Integrating both sides:$\int \cot v \, dv = \int \frac{dx}{x}$$\log|\sin v| = \log|x| + \log|c|$$\log|\sin v| = \log|cx|$$\sin v = cx$Substituting $v = \frac{y}{x}$ back:$\sin(\frac{y}{x}) = cx$

If $\frac{dy}{dx} = \frac{y + x \tan(\frac{y}{x})}{x}$, then $\sin(\frac{y}{x})$ is equal to

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