find the particular solution satisfying the given condition, for the following differential equation: `(x+1)(dy)/(dx)=2e^-y-1` given that `y=0` when `x=0`

Explanation

Given differential equation is
`(x+1)(dy)/(dx)=2e^(-y)-1`……`(1)`
`implies (dy)/(2e^(-y)-1)=(dx)/(x+1)implies(e^(y)dy)/(2-e^(y))=(dx)/(x+1)`
On integration, `int(e^(y)dy)/(2-e^(y))=int(dx)/(x+1)`……`(2)`
Let `2-e^(y)=timplies-e^(y)=(dt)/(dy)impliese^(y)dy=-dt`
From equation `(2)`, `int-(dt)/(t)=int(dx)/(x+1)`
`implies -log|t|=log|x+1|+logC`
`implies -log|2-e^(y)|=log|C(x+1)|`
`implies (1)/(2-e^(y))=C(x+1)`
`implies 2-e^(y)=(1)/(C(x+1))`
Now, at `x=0` and `y=0`, `2-1=(1)/(C)impliesC=1`
Put the value of `C` in equation `(3)`,
`2-e^(y)=(1)/((x+1))impliese^(y)=2-(1)/(x+1)`
`impliese^(y)=(2x+2-1)/(x+1)impliese^(y)=(2x+1)/(x+1)`
`implies y=log|(2x+1)/(x+1)|`