Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(3 y^{2}-5 x^{2}\right) y dx+2 x\left(x^{2}-y^{2}\right) dy=0$ such that $\mathrm{y}(1)=1$. then $\left|(\mathrm{y}(2))^{3}-12 \mathrm{y}(2)\right|$ is equal to:

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Differential Equations

Let

y = y (x)

be the solution of the differential equation

(3 y^{2} - 5 x^{2}) y d x + 2 x (x^{2} - y^{2}) d y = 0

such that

y (1) = 1

. then

(y (2))^{3} - 12 y (2)

is equal to:

Answer: (A)

The equation is homogeneous; with

y = v x

it separates to

\frac{d x}{x} = \frac{2 ( v ^{2} - 1 )}{v ( v ^{2} - 3 )} d v .

Integrating,

ln ∣ x ∣ = \frac{2}{3} ln v (v^{2} - 3) + C

, and

y (1) = 1

(so

v = 1

) fixes

C = - \frac{2}{3} ln 2.

Thus

v (v^{2} - 3) = 2 x^{3/2}

, and since

v (v^{2} - 3) = \frac{y ( y ^{2} - 3 x ^{2} )}{x ^{3}}

y (y^{2} - 3 x^{2}) = 2 x^{9/2} .

x = 2

this is

2 \cdot 2^{9/2} = 322

, so

y (2)^{3} - 12 y (2) = 322 .

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Single correct option

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