Question

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`

Answer 1

Correct Answer - ` y = log |( 2x + 1 )/( x +1 )|`
` (x + 1) ( dy )/(dx) = (( 2 - e ^(y)))/( e ^(y)) rArr (dx)/(dy) = ((x + 1 )e ^(y))/(( 2 -e ^(y))`
`therefore (dx)/(dy) - (e ^(y))/( ( 2 - e ^(y))) *x = (e ^(y))/(( 2 - e ^(y)))`.
`IF = e^(int ( - e ^(y))/(( 2- e ^(y))dy ) = e ^( log ( 2 - e ^(y))) = ( 2- e ^(y))`
` therefore x xx ( 2 - e ^(y)) = int (e^(y))/( ( 2- e ^(y))) xx ( 2- e ^(y)) dy + C = int ( e ^(y) dy +C = ey +C `.
Now`, y =0 and x = 0 rArr C = - 1 `
` therefore x ( 2 - e^(y)) = e ^(y) - 1 rArr ( 2x + 1 ) = (x + 1 ) e ^(y)`
`therefore e ^(y) = ((2 x + 1 ))/((x + 1 )) rArr y = log (( 2x + 1 )/(x + 1 )) `