Let y = y x be the solution of the differential equation x + 1 y ' - y = e 3 x x + 1 2 , with y 0 = 1 3 . Then, the point x = - 4 3 for the curve y = y x is

Step-by-step solution

x + 1 d y - y d x = e 3 x x + 1 2 d x x + 1 d y - y d x ( x + 1 ) 2 = e 3 x d x d y x + 1 = e 3 x d x On integration, we get y x + 1 = e 3 x 3 + C Given y 0 = 1 3 So C = 0 ⇒ y = x + 1 e 3 x 3 d y d x = e 3 x 3 3 x + 4 d 2 y d x 2 = e 3 x 3 x + 5 Clearly, x = - 4 3 is a point of local minima as d 2 y d x 2 x = - 4 3 < 0