I: y^{prime}=frac{y+x}{x} ; quad II: y^{prime | vedclass

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(C) differential equation $\frac{dy}{dx} = f(x, y)$ is homogeneous if $f(x, y)$ is a homogeneous function of degree $0$.For $I: f(x, y) = \frac{y+x}{x

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only $S3$ is valid

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(C) differential equation $\frac{dy}{dx} = f(x, y)$ is homogeneous if $f(x, y)$ is a homogeneous function of degree $0$.For $I: f(x, y) = \frac{y+x}{x} = \frac{y}{x} + 1$. This is a homogeneous function of degree $0$. Thus,$I$ is a homogeneous differential equation.For $II: f(x, y) = \frac{x^2+y}{x^3} = \frac{1}{x} + \frac{y}{x^3}$. This is not a homogeneous function because the degrees of the terms are not the same. Thus,$II$ is not homogeneous.For $III: f(x, y) = \frac{2xy}{y^2-x^2}$. Dividing numerator and denominator by $x^2$,we get $f(x, y) = \frac{2(y/x)}{(y/x)^2-1}$. This is a homogeneous function of degree $0$. Thus,$III$ is a homogeneous differential equation.Since $I$ and $III$ are homogeneous,statement $S3$ is valid.

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