$I: y^{\prime}=\frac{y+x}{x} ; \quad \text { II }: y^{\prime}=\frac{x^2+y}{x^3} ; \quad \text { III }:…

$I: y^{\prime}=\frac{y+x}{x} ; \quad \text { II }: y^{\prime}=\frac{x^2+y}{x^3} ; \quad \text { III }: y^{\prime}=\frac{2 x y}{y^2-x^2}$ S1: Differential equations given by I and II are homogeneous differential equations. S2: Differential equations given by II and III are homogeneous differential equations. S3: Differential equations given by I and III are homogeneous differential equations.
  1. only $\mathrm{S} 1$ is valid
  2. both S1 and S2 are valid
  3. only S3 is valid
  4. only S2 is valid

Solution

We will check all 3 equations. I : $\frac{d y}{d x}=\frac{y+x}{x} \Rightarrow$ All terms have same degree equal to 1 . II : $\frac{d y}{d x}=\frac{x^2+y}{y^3} \Rightarrow$ Here all terms have different degrees., III : $\frac{d y}{d x}=\frac{2 x y}{y^2-x^2} \Rightarrow$ All terms have same degree equal to 2 .

Asked in: MHT CET 2021 (21 Sep Shift 2)

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