CUET · MATHS · PYQ PAPER 2025

Which one of the following equations is a homogeneous differential equation?

A \((4x + 5) dy + (3y - 4) dx = 0\)
B \(x^2 y d x-\left(x^3+y^3\right) d y=0\)
C \(\left(x^3+2 y^2 x\right) d y+2 x y d x=0\)
D \(x^2 d y-y^2 d x=\sqrt{x^2+y^2} d x\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(x^2 y d x-\left(x^3+y^3\right) d y=0\)

Step-by-step Solution

Detailed explanation

\(M(x, y) = x^2 y\) \(M(tx, ty) = (tx)^2(ty) = t^3 x^2 y\) \(N(x, y) = -(x^3+y^3)\) \(N(tx, ty) = -((tx)^3+(ty)^3) = -t^3(x^3+y^3)\) Both \(M(x, y)\) and \(N(x, y)\) are homogeneous functions of degree 3. Therefore, \(x^2 y d x-\left(x^3+y^3\right) d y=0\) is a homogeneous…

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.

Same subject

For \(\ x \in R -\{-1,0,1\} \), \(\int \frac{1}{x-x^5} d x\) is equal to:

CUET 2025 Easy

Match List I with List II

LIST I	LIST II
A. If \(4 \sin ^{-1} x+\cos ^{-1} x=\pi\), then \(x\) equal to	I. \(\frac{\pi}{2}\)
B. The value of \(\frac{1-\tan ^2 15^{\circ}}{1+\tan ^2 15^{\circ}}\) is	II. \(\frac{1}{2}\)
C. If \(x+\frac{1}{x}=2\), then principal value of \(\sin ^{-1} x\) is	III. \(\frac{3 \pi}{4}\)
D. Two angles of a triangle are \(\cot ^{-1} 2\) and \(\cot ^{-1} 3\), then third angle is	IV. \(\frac{\sqrt{3}}{2}\)

CUET 2023 Easy

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