Let y ( x ) be a solution of the differential equation 1 + e x   y ′ + y e x = 1 . If y 0 = 2 ,…

Let y(x) be a solution of the differential equation 1+ex y′+yex=1. If y0=2 , then which of the following statements is (are) true?
  1. y-4=0
  2. y-2=0
  3. y(x) has a critical point in the interval (-1, 0)
  4. y(x) has no critical point in the interval (-1, 0)

Solution

1+exdy+yexdx=dx
dy+dex.y=dx
yex=x-y+c
Now, x=0, y=2
2.e0=0-2+c
⇒c=4
Now, yex=x-y+4
y=x+4ex+1
Now, dydx=ex+1-x+4exex+1
=ex+1-xex-4exex+1=1-xex-3exex+1
f′0=-ve
f′-1=1+e-3e-1e-1+1=+ve
Critical point in -1, 0

Asked in: JEE Advanced 2015 (Paper 1)

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