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

Options

A. `y(-4)=0`
B. `y(-2)=0`
C. `y(x)` has 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))y'+ye^(x)=1`
`rArr" "y'+(e^(x))/(1+e^(x))y=(1)/(1+e^(x))" ...(i)"`
This is a linear differential equation with integrating factor
`"I.F."=e^(int(e^(x))/(1+e^(x))dx)=e^(log(1+e^(x)))=1+e^(x)`
Multiplying both sides of (i) by I.F. `=1+e^(x)`, we obtain
`y'(1+e^(x))+ye^(x)=1`
Integrating both sides with respect to x, we obtain
`y(1+e^(x))=x+C" ...(ii)"`
It is givne that `y(0)=2` i.e.y = 2 when x = 0. Putting x = 0, y= 2 in (ii). we get
`4=0 +C rArr C=4`
Putting C = 4 in (ii) we obtain
`y(1+e^(x))=x+4rArry(x+4)/(1+e^(x))rArry(-4)=0`
Putting `y=(x+4)/(1+e^(x))` in (i), we obtain
`y'=((1+e^(x))-(x+4)e^(x))/((1+e^(x))^(2))`
`rArry'(-1)((1+e^(-1))-3e^(-1))/((1+e^(-1))^(2))=(e-2)/(e(1+e^(-1))^(2))gt0 and y'(0)=-(1)/(2)lt0`
`rArr y'(x)=0" for some x" in(-1,0)`
`rArry(x)" has a critical point in the interval"(-1,0).`
Hence, options (a)and (c) are true.