Let y=y(x) be the solution curve of the differential equation (d y)/(d x)+1/(x²-1) y=((x-1)/(x+1))^(1 / 2), x >1 passing through the point (2, √(1/3)). Then √7 y(8) is equal to :

Solution

Maths sir · 4 steps

The integrating factor √((x-1)/(x+1)) makes the right side easy to integrate.

1. Integrating factor. Here P(x) = 1/(x²-1), and ∫ dx/(x²-1) = 1/2 ln(x-1)/(x+1). So the integrating factor is e^(1/2 ln(x-1)/(x+1)) = √((x-1)/(x+1)). This is valid because x > 1.

   y√((x-1)/(x+1)) = ∫ (x-1)/(x+1) dx = ∫(1 - 2/(x+1))dx = x - 2 ln(x+1) + C

2. Find C. At x = 2 the factor is 1/√3 and y = 1/√3, so the left side is 1/3. Then 1/3 = 2 - 2 ln 3 + C, which gives C = 2 ln 3 - 5/3.
3. Put x = 8. The factor is √(7/9) = √7/3. So √7/3 y(8) = 8 - 2 ln 9 + 2 ln 3 - 5/3 = 19/3 - 2 ln 3, using ln 9 = 2 ln 3.
4. So. Multiplying by 3 gives √7 y(8) = 19 - 6 logₑ 3, which is option (d).

The idea to keep. For dy/dx + y/(x²-1), the integrating factor is √((x-1)/(x+1)). It is often built to cancel the messy root on the right, so check for that first.

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