Let y = y x , x > 1 , be the solution of the differential equation x - 1 d y d x + 2 x y = 1 x - 1 , with y 2 = 1 + e 4 2 e 4 . If y 3 = e α + 1 β e α . then the value of α + β is equal to ______.

Step-by-step solution

Given, x - 1 d y d x + 2 x y = 1 x - 1 ⇒ d y d x + 2 x x - 1 · y = 1 x - 1 2 I . F = e ∫ 2 x x - 1 = e 2 x × x - 1 2 Solution will be, y × I . F = ∫ 1 x - 1 2 × I . F Solving above with y 2 = 1 + e 4 2 e 4 we get, y = 1 x - 1 2 e 2 x + 1 2 e 2 x So, y 3 = e 6 + 1 8 e 6 , now on comparing with y 3 = e α + 1 β e α we get, α + β = 14