The solution of `(dy)/(dx)-y=1, y(0)=1` is given by

Options

A. `xy=-e^(x)`
B. `xy=-e^(-x)`
C. `xy=-1`
D. `y=2e^(x)-1`

Answer

D

Explanation

Given that,
`" "(dy)/(dx)-y=1`
`rArr" "(dy)/(dx)=1+y`
`rArr" "(dy)/(1+y)=dx`
On integrating both sides, we get
`" "log(1+y)=x+C" "`...(i)
when x=0 and y=1, then
`" "log2=0+c`
`rArr" "C=log2`
The required solution is
`" "log(1+y)=x+log2`
`rArr" "log((1+y)/(2))=x`
`rArr" "(1+y)/(2)=e^(x)`
`rArr" "1+y=2e^(x)` k
`rArr" "y=2e^(x)-1`