Let y = y(x), x > 1, be the solution of the differential equation (x - 1)dy/dx + 2xy = 1/(x - 1), with y(2) = (1 + e⁴)/2e⁴. If y(3) = (e^α + 1)/(β e^α), then the value of α + β is equal to _________.

Solution

Maths sir · 5 steps

Divide by (x-1), then split 2x/(x-1) as 2 + 2/(x-1) for a clean integrating factor.

1. Standard form. Divide through by (x-1) to get dy/dx + 2x/(x-1) y = 1/(x-1)². Write 2x/(x-1) = 2 + 2/(x-1), so P = 2 + 2/(x-1).
2. Integrating factor. ∫ P dx = 2x + 2 ln(x-1), so the integrating factor is e²ˣ(x-1)². The (x-1)² on the right side cancels against it.

   y e²ˣ(x-1)² = ∫ e²ˣ dx = e²ˣ/2 + C

3. Use y(2). At x = 2 the left side is y(2) e⁴ = (1+e⁴)/2. This equals e⁴/2 + C, so C = 1/2.
4. Find y(3). At x = 3: 4e⁶ y(3) = (e⁶ + 1)/2, so y(3) = (e⁶ + 1)/8e⁶.
5. So. Matching with (e^α + 1)/(β e^α) gives α = 6 and β = 8, so α + β = 14.

The idea to keep. In a linear equation, always get dy/dx alone before reading off P. Then break P into simple pieces, because the integrating factor is often built to cancel the right side.

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