Let x = x ( t ) and y = y ( t ) be solutions of the differential equations dx dt + ax = 0 and dy dt + by = 0 respectively, a , b ∈ R . Given that x ( 0 ) = 2 ; y ( 0 ) = 1 and 3 y ( 1 ) = 2 x ( 1 ) , the value of t , for which x ( t ) = y ( t ) , is :

Step-by-step solution

Given: dx dt + ax = 0 ⇒ dx x = - adt Integrating both sides, ⇒ ∫ dx x = - a ∫ dt ⇒ log | x | = - at + c At t = 0 , x = 2 ⇒ log 2 = 0 + c ⇒ logx = - at + log 2 ⇒ logx - log 2 = - at ⇒ log x 2 = - at ⇒ x 2 = e - at ⇒ x = 2 e - at . . . ( i ) It is also given that, dy dt + by = 0 ⇒ dy y = - bdt ⇒ log | y | = - bt + λ At, t = 0 , y = 1 ⇒ 0 = 0 + λ ⇒ y = e - bt . . . ( ii ) According to question, 3 y ( 1 ) = 2 x ( 1 ) ⇒ 3 e - b = 2 2 e - a ⇒ e a - b = 4 3 For x t = y t ⇒ 2 e - at = e - bt ⇒ 2 = e ( a - b ) t ⇒ 2 = 4 3 t