MHT CET · Maths · Differential Equations

The differential equation \(x \frac{\mathrm{dy}}{\mathrm{d} x}=2 \mathrm{y}\) represents ....

A a family of circles with radius \(c\).
B a family of parabolas with vertex at the origin and axis along the positive Y-axis
C a family of parabolas with vertex the at origin and axis along the positive X-axis
D a family of ellipses

Verified Solution

Answer & Solution

Correct Answer

(B) a family of parabolas with vertex at the origin and axis along the positive Y-axis

Step-by-step Solution

Detailed explanation

\(\frac{\mathrm{dy}}{\mathrm{y}} = \frac{2}{\mathrm{x}} \mathrm{dx}\) \(\int \frac{1}{\mathrm{y}} \mathrm{dy} = \int \frac{2}{\mathrm{x}} \mathrm{dx}\)

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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