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

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(B) Given the differential equation: $\frac{dy}{dx} = \frac{y + x 	an(\frac{y}{x})}{x} \quad (i)$ This is a homogeneous differential equation. Let $y

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$cx$

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(B) Given the differential equation: $\frac{dy}{dx} = \frac{y + x 	an(\frac{y}{x})}{x} \quad (i)$ This is a homogeneous differential equation. Let $y = vx$,then differentiating with respect to $x$,we get $\frac{dy}{dx} = v + x \frac{dv}{dx}$. Substituting these into Eq. $(i)$: $v + x \frac{dv}{dx} = \frac{vx + x 	an(v)}{x}$ $v + x \frac{dv}{dx} = v + 	an(v)$ $x \frac{dv}{dx} = 	an(v)$ Separating the variables,we get: $\cot(v) dv = \frac{1}{x} dx$ Integrating both sides: $\int \cot(v) dv = \int \frac{1}{x} dx$ $\log|\sin(v)| = \log|x| + \log|c|$ $\log|\sin(v)| = \log|cx|$ Taking the exponential of both sides: $\sin(v) = cx$ 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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