JEE 2023 — Mathematics PYQ

Step 1: Equation ko simplify karein

Di gayi equation:

Kyunki har term ki degree 3 hai, yeh ek homogeneous differential equation hai.

Step 2: Substitution $y=vx$ rakhein

Differentiate karne par: $\frac{dy}{dx}=v+x\frac{dv}{dx}$

Equation mein values rakhne par:

Step 3: Variables separate karke Integrate karein

LHS ko solve karne ke liye, dhyaan dein ki $d(3v-v^3)=(3-3v^2)dv$.

Hum LHS ko likh sakte hain: $\frac{2}{3}\int \frac{3-3v^2}{3v-v^3}dv$

$v=y/x$ wapas rakhne par:

Step 4: Boundary condition $y(1)=1$ apply karein

Humein equation mili: $(3y{x}^2-y^3{)}^2=4x^9$

Dono taraf square root lene par: $3y{x}^2-y^3=\pm 2x^{9/2}$

Step 5: $y(2)$ ki value find karein

$x=2$ rakhne par:

Humein $|(y(2){)}^3-12y(2)|$ ki value nikaalni hai:

Correct Option: (2) $32\sqrt{2}$