Let y=y(x) be the solution of the differential equation
sec ² x d x+(e²ʸ tan ² x+tan x) d y=0,0< x<π/2, y(π / 4)=0.
If y(π / 6)=α, then e^(8 α) is equal to ____________.

Solution

Maths sir · 4 steps

Put t = tan x and treat t as the function of y: the equation becomes Bernoulli in t.

1. Change of variable. With t = tan x, dt = sec² x dx, so the equation becomes dt + (e²ʸt² + t)dy = 0. Taking t as the function of y gives dt/dy + t = -e²ʸt².
2. Make it linear. Divide by t² and put u = 1/t = cot x. Then du/dy = -1/t²dt/dy, and the equation becomes linear in u.

   du/dy - u = e²ʸ

3. Integrating factor. Multiply by e⁻ʸ to get d/dy(u e⁻ʸ) = eʸ, so u e⁻ʸ = eʸ + C. At x = π/4, y = 0 and u = cot π/4 = 1, so C = 0 and cot x = e²ʸ.
4. So. At x = π/6, e^(2α) = cot π/6 = √3. Then e^(8α) = (e^(2α))⁴ = (√3)⁴, which is 9.

The idea to keep. When dy carries the messy terms, flip roles and treat x (or tan x) as the function of y. A squared term on the right means Bernoulli: divide by it and put u = 1/t.

Answer from the NTA final answer key. Solution by PrepWiser.