If y=y(x) is the solution of the differential equation
(d y)/(d x)+(4 x)/(x²-1) y=(x+2)/(x²-1)^(5/2), x > 1 such that
y(2)=2/9 log ₑ(2+√3) and y(√2)=α log ₑ(√α+β)+β-√γ, α, β, γ ∈ ℕ , then α β γ is equal to :

Solution

Maths sir · 5 steps

The integrating factor (x²-1)² eats most of the right side; what is left integrates in one line.

1. Integrating factor. With P = 4x/(x²-1), we get ∫ P dx = 2 ln(x²-1). So the integrating factor is e^(2 ln(x²-1)) = (x²-1)².

   y (x²-1)² = ∫ (x+2)/√(x²-1) dx = √(x²-1) + 2 ln(x+√(x²-1)) + C

2. Split the integral. The part ∫ x/√(x²-1) dx gives √(x²-1). The part ∫ 2/√(x²-1) dx gives 2 ln(x+√(x²-1)).
3. Find C. At x = 2 the left side is 9y(2) = 2 ln(2+√3). The right side is √3 + 2 ln(2+√3) + C. So C = -√3.
4. Value at root 2. At x = √2, (x²-1)² = 1 and √(x²-1) = 1. So y(√2) = 2 ln(√2+1) + 1 - √3.
5. So. Matching this with α ln(√α+β) + β - √γ gives α = 2, β = 1 and γ = 3. Hence αβγ = 6.

The idea to keep. When P(x) is k times the derivative of something over that something, the integrating factor is that something to the power k. Then fix C from the given value before you substitute anything else.

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