CUET UG 2025 — Mathematics Calculus

Order $=$= the highest derivative in the equation.

Degree $=$= the power of the highest derivative (equation must be polynomial in derivatives).

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Given equation:

$$2x^3\frac{dy}{dx}-5\left(\frac{d^2y}{dx^2}\right)^2=6\left(\frac{dy}{dx}\right)^3$$2x3dxdy​−5(dx2d2y​)2=6(dxdy​)3

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$\frac{dy}{dx}$dxdy​ is the first derivative

$\frac{d^2y}{dx^2}$dx2d2y​ is the second derivative (highest)

Therefore, order $m = 2$m=2

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The equation is already polynomial in derivatives. The highest order derivative is $\frac{d^2y}{dx^2}$dx2d2y​, and it appears as:

$$\left(\frac{d^2y}{dx^2}\right)^{2}$$(dx2d2y​)2

Therefore, degree $n = 2$n=2

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$(A)$(A) $m = 2$m=2 ✓

$(B)$(B) $n = 3$n=3 ✗ (since $n = 2$n=2)

$(C)$(C) $m = 3$m=3 ✗ (since $m = 2$m=2)

$(D)$(D) $mn = 4$mn=4 ✓ (since $m \times n = 2 \times 2 = 4$m×n=2×2=4)

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Options $(A)$(A) and $(D)$(D) are correct.