If dy/dx + 2ˣ⁻ʸ(2ʸ - 1)/(2ˣ - 1) = 0, x, y > 0, y(1) = 1, then y(2) is equal to :

Solution

Maths sir · 4 steps

Split 2ˣ⁻ʸ into 2ˣ/2ʸ and the equation separates at once, x on one side and y on the other.

1. Separate variables. Write 2ˣ⁻ʸ = 2ˣ/2ʸ. Then dy/dx = -2ˣ/(2ˣ - 1) · (2ʸ - 1)/2ʸ, so move every y-term to the left and every x-term to the right.

   ∫ 2ʸ/(2ʸ - 1) dy = -∫ 2ˣ/(2ˣ - 1) dx

2. Integrate both sides. The derivative of 2ʸ - 1 is 2ʸ ln 2, so the left side is ln(2ʸ - 1)/(ln 2). The right side is -ln(2ˣ - 1)/(ln 2) in the same way. Multiply through by ln 2 and combine the logs to get (2ʸ - 1)(2ˣ - 1) = K.
3. Use y(1) = 1. At x = 1, y = 1: (2 - 1)(2 - 1) = 1, so K = 1 and the solution is (2ʸ - 1)(2ˣ - 1) = 1.
4. So. At x = 2: 3(2ʸ - 1) = 1, so 2ʸ = 4/3. Then y = log₂ 4 - log₂ 3 = 2 - log₂ 3, which is option (d).

The idea to keep. When the equation has aˣ⁻ʸ, split it into aˣ/aʸ first. Then look for the form f'/f: each side integrates to a log, and the logs combine into a product equal to a constant.

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