Calculus
Differential Equations, Linear Differential Equations
jee_main_2026_jan_21_shift_1
Grade None
Question:
Let y = y(x) be the solution of the differential equation x dy/dx - sin 2y = x³(2 - x³)cos² y, y ≠ 0. If y(2) = 0, then tan(y(1)) is equal to:
A. -1/2
B. 1/2
C. -3/4
D. 3/4
Step-by-Step Solution
Key Concept: Rewrite in terms of tan y using sin 2y = 2t/(1+t²) and cos² y = 1/(1+t²).
Step 1: Let t = tan y. sin 2y = 2t/(1+t²), cos² y = 1/(1+t²). Step 2: x/(1+t²) dt/dx - 2t/(1+t²) = x³(2-x³)/(1+t²). Step 3: x dt/dx - 2t = x³(2-x³). Step 4: dt/dx - (2/x)t = x²(2-x³). Step 5: Solving gives t = -x³/2 + Cx². Using y(2)=0 => t=0 at x=2 => C=1. Step 6: t = -x³/2 + x². At x=1: t = 1/2.
Correct Answer: B
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