MHT CET · Maths · Differential Equations

\(I: y^{\prime}=\frac{y+x}{x} ; \quad \text { II }: y^{\prime}=\frac{x^2+y}{x^3} ; \quad \text { III }: y^{\prime}=\frac{2 x y}{y^2-x^2}\)
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.

A only \(\mathrm{S} 1\) is valid
B both S1 and S2 are valid
C only S3 is valid
D only S2 is valid

Verified Solution

Answer & Solution

Correct Answer

(C) only S3 is valid

Step-by-step Solution

Detailed explanation

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 .

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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