JEE Main 2024 Mathematics question
Let y = y(x) be the solution of the differential equation sec²x dx + (e²ʸ tan²x + tan x)dy = 0, 0 < x < π/2, y(π/4) = 0. If y(π/6) = α, then e⁸ᵅ is equal to .
Answer: 9
Solution
Put tan x = t: dt/dy + e²ʸt² + t = 0, i.e. (1/t²)(dt/dy) + 1/t = −e²ʸ. With u = 1/t: du/dy − u = e²ʸ, with integrating factor e⁻ʸ: u e⁻ʸ
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