Let the solution curve x=x(y), 0 < y < π/2, of the differential equation (log ₑ(cos y))² cos y d x-(1+3 x log ₑ(cos y)) sin y d y=0 satisfy x(π/3)=1/(2 log ₑ 2). If x(π/6)=1/(log ₑ m-log ₑ n), where m and n are coprime, then m n is equal to __________.

Solution

Maths sir · 5 steps

Put t = ln(cos y): then tan y dy = -dt and the equation becomes linear in x.

1. Change variable. Divide by (ln cos y)²cos y to get dx = ((1 + 3x ln cos y)tan y)/(ln cos y)² dy. Let t = ln(cos y), so dt = -tan y dy. The equation becomes dx = -(1 + 3xt)/t² dt.

   dx/dt + 3/t x = -1/t²

2. Integrating factor. This is linear in x, with integrating factor e^(∫ 3/t dt) = t³. So d/dt(x t³) = -t, which gives x t³ = -t²/2 + C.
3. Find C. At y = π/3, t = ln 1/2 = -ln 2 and x = 1/(2 ln 2). The left side is 1/(2 ln 2) × (-(ln 2)³) = -(ln 2)²/2, which already equals -t²/2. So C = 0 and x = -1/2t.
4. Evaluate at π/6. At y = π/6, t = ln√3/2 = 1/2 ln 3 - ln 2, so 2t = ln 3 - ln 4. Then x = -1/(ln 3 - ln 4) = 1/(ln 4 - ln 3).
5. So. m = 4 and n = 3, which are coprime. So mn = 12.

The idea to keep. When ln(cos y) and tan y dy both appear, substitute t = ln(cos y). Also, when x is the unknown function of y, look for a linear equation in x before trying to separate variables.

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