For x ∈ ℝ , let y x be a solution of the differential equation x 2 - 5 d y d x - 2 x y = - 2 x x 2 - 5 2 such that y 2 = 7 . Then the maximum value of the function y x is

Step-by-step solution

Given, x 2 - 5 d y d x - 2 x y = - 2 x x 2 - 5 2 ⇒ d y d x + - 2 x x 2 - 5 y = - 2 x x 2 - 5 Which is a linear differential equation, So, integrating factor,  I.F. = e ∫ - 2 x x 2 - 5 d x = 1 x 2 − 5 Now solution of differential equation is given by, y ⋅ 1 x 2 − 5 = ∫ − 2 x ⋅ x 2 − 5 x 2 − 5 d x ⇒ y x 2 − 5 = x 2 − 5 x 2 − 5 − x 2 + C Now using the given value, y ( 2 ) = 7 ⇒ C = 3 ⇒ y = - x 2 x 2 - 5 + 3 x 2 -