What is the solution of `(dy)/(dx) + 2y =1` satisfying x=0,y=0 ?

Options

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

Answer

A

Explanation

` (dy)/(dx) +2y = 1 Rightarrow (dy)/(dx) =1 -2y`
` int (dy)/(1-2y) int dx`
` -1/2 log |1 -2y|=x +C`
at x =0, y =0
` -1/2 log 1 =0 +C Rightarrow C=0`
` 1-2y = e^(-2x)`
` y = (1-e^(-2x))/2`