Let y = y(x) be the solution of the differential equation dy/dx + (2 y)/(2 cos ⁴x - cos ²x) = x e^(tan ⁻¹(2 cot 2x)), 0 < x < (π)/2 with y((π)/4) = (π ²)/32. If y((π)/3) = (π ²)/18e^(- tan ⁻¹(α)), then the value of 3 α ² is equal to _ _ _ _ _ _.

Differential Equations: JEE Main 2022 Mathematics question

JEE Main 2022 · 29 Jun, Morning shift · Q36

JEE Main 202229 Jun, Morning shiftMathematicsHardDifferential Equations163× since 2002

Let y = y(x) be the solution of the differential equation

{{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 < x < {\pi \over 2}

with

y\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {32}}

. If

y\left( {{\pi \over 3}} \right) = {{{\pi ^2}} \over {18}}{e^{ - {{\tan }^{ - 1}}(\alpha )}}

, then the value of 3

\alpha

² is equal to ___________.

Your answerType your answer. Integers and decimals are accepted.