CUET UG 2025 — Mathematics Calculus

The degree of a differential equation is the highest power of the highest order derivative when the equation is in polynomial form (no roots, no fractions with derivatives).

Given equation:

$$\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}} = a^2 \frac{d^2y}{dx^2}$$(2+(dxdy​)2)23​=a2dx2d2y​

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The highest order derivative in the equation is $\frac{d^2y}{dx^2}$dx2d2y​ (second order).

The left side has a fractional power $\frac{3}{2}$23​, so the equation is not in polynomial form.

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Squaring both sides:

$$\left[\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}}\right]^{2} = \left[a^2 \frac{d^2y}{dx^2}\right]^{2}$$​(2+(dxdy​)2)23​​2=[a2dx2d2y​]2

$$\left(2 + \left(\frac{dy}{dx}\right)^2\right)^3 = a^4 \left(\frac{d^2y}{dx^2}\right)^{2}$$(2+(dxdy​)2)3=a4(dx2d2y​)2

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The equation is now in polynomial form.

The highest order derivative $\frac{d^2y}{dx^2}$dx2d2y​ appears with power $2$2.

Therefore, the degree of the differential equation is $2$2.