Question

Let y(x) be a solution of the differential equation (1 + ex)y' + ye^x = 1. If y(0) = 2, which of the following statements (is/are) true?

(a) y(– 4) = 0 

(b) y(2) = 0 

(c) y(x) has a critical point in (–1, 0) 

(d) y(x) has no critical point in (–1, 0).

Answer 1

Answer is (c) y(x) has a critical point in (–1, 0)

Explanation:

We have (1 + e x )y' + ye^x = 1

Thus, y (–4) = 0

Clearly, g(0) = (2 – 4)/2² < 0

and

g(0).g(– 1) < 0 

Hence, 

g(x) has a root in between (– 1, 0).