If y = y(x) is the solution of the equation
eˢⁱⁿʸ cos ydy/dx + eˢⁱⁿʸ cos x = cos x, y(0) = 0; then
1 + y( π/6 ) + √3/2y( π/3 ) + 1/√2y( π/4 ) is equal to ____________.

Solution

Maths sir · 4 steps

Put t = eˢⁱⁿʸ and the equation turns linear in t. Then the initial condition forces y to be 0.

1. Spot the substitution. Let t = eˢⁱⁿʸ. By the chain rule, dt/dx = eˢⁱⁿʸ cos y dy/dx, which is exactly the first term. So the equation becomes linear in t.

   dt/dx + t cos x = cos x

2. Integrating factor. The integrating factor is e^(∫ cos x dx) = eˢⁱⁿˣ. So t eˢⁱⁿˣ = ∫ cos x eˢⁱⁿˣ dx = eˢⁱⁿˣ + C.
3. Use y(0) = 0. At x = 0, t = eˢⁱⁿ⁰ = 1 and eˢⁱⁿ⁰ = 1. This gives 1 = 1 + C, so C = 0. Then t = 1 for every x, so eˢⁱⁿʸ = 1 and sin y = 0. Since y starts at 0 and changes continuously, y(x) = 0 throughout.
4. So. y(π/6), y(π/3) and y(π/4) are all 0. The expression is 1 + 0 + 0 + 0 = 1.

The idea to keep. When a function of y and its derivative f'(y)dy/dx both appear, substitute t = f(y). The equation usually becomes linear in t, and you can solve it with an integrating factor.

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