Let `y(x)` be a solution of the differential equation `(1+e^x)y^(prime)+y e^x=1.` If `y(0)=2` , then which of the following statements is (are) true? (a) `( b ) (c) y(( d ) (e)-4( f ))=0( g )` (h) (b) `( i ) (j) y(( k ) (l)-2( m ))=0( n )` (o) (c) `( d ) (e) y(( f ) x (g))( h )` (i) has a critical point in the interval `( j ) (k)(( l ) (m)-1,0( n ))( o )` (p) (q) `( r ) (s) y(( t ) x (u))( v )` (w) has no critical point in the interval `( x ) (y)(( z ) (aa)-1,0( b b ))( c c )` (dd)

Options

A. `y(-4)=0`
B. `y(-2)=0`
C. `y(x)` has a critical point in the interval `(-1,0)`
D. `y(x)` has no critical point in the interval `(-1,0)`

Answer

A, C

Explanation

We have `(1+e^(x))(dy)/(dx)+ye^(x)=1`
`rArr (dy)/(dx) + e^(x)/(1+e^(x))y=1/(1+e^(x))` (Linear differential equation)
`I.F. =e^(int(e_(x)/(1+e^(x))))=1+e^(x)`
Therefore, solution is
`y/(1+e^(x))y=x+c`
or `(1+e^(x))y=x+c`
Given, `x=0, y=2,` so `c=4`
`therefore y=(x+4)/(e^(x)+1)`
`y(-4)=0`
`y(-2)=2/(e^(-2)+1)`
`(dy)/(dx) = ((e^(x)+1).1-(x+4)e^(x))/(e^(x)+1)^(2)`
`=(e^(x)(-x-3)+1)/(e^(x)+1)^(2)`
When `(dy)/(dx)=0`, then `x+3=e^(-x)`
Graphs of `y=x+3` and `y=e^(-x)` are as shown in the following figure.

Clearly, graphs intersect for `x in (-1,0`0
Hence, `y(x)` has a critical point in the interval `(-1,0)`