If x = x t is the solution of the differential equation t + 1 d x = 2 x + t + 1 4 d t , x 0 = 2 , then x 1 equals ________

Step-by-step solution

Given: t + 1 d x = 2 x + t + 1 4 d t ⇒ d x d t = 2 x + t + 1 4 t + 1 ⇒ d x d t - 2 x t + 1 = t + 1 3 ⇒ IF = e ∫ - 2 t + 1 d t ⇒ IF = e - 2 log t + 1 ⇒ IF = 1 t + 1 2 Solution of the differential equations is given by, ⇒ x × 1 t + 1 2 = ∫ t + 1 d t ⇒ x t + 1 2 = t 2 2 + t + c It is given that x 0 = 2 ⇒ 2 0 + 1 2 = 0 2 + 0 + c ⇒ c = 2 ⇒ x t + 1 2 = t 2 2 + t + 2 Putting t = 1 , ⇒ x 1 + 1 2 = 1 2 2 + 1 + 2 ⇒ x 4 = 1 2 + 3 ⇒ x = 2 + 12 ⇒ x = 14