Find the general solution of the differential equation
`y-x(dy)/(dx)=a(y^(2)+(dy)/(dx))`.

Options

A. `(x+a) (x+ay)=cy`
B. `(x+a)(1-ay)=cy`
C. `(x+a)(1-ay)=c`
D. None of these

Answer

B

Explanation

Given `, y - x ( dy ) /( dx ) = a (y ^ 2 + ( dy ) /( dx )) `
` rArr y - ay ^ 2 = a ( dy ) /( dx ) + x ( dy ) /( dx ) `
` rArr y ( 1 - ay ) = ( a + x ) ( dy ) /( dx ) `
` rArr (dx ) / ( a + x ) = ( dy ) / ( y ( 1 - ay )) `
on integrating both sides, we get
` log ( a + x ) = int [ ( 1 ) /( y ) + ( a ) /( 1 - ay )] dx `
` rArr log ( a + x ) = log y + ( along ( 1 - ay ))/( -a) + log c `
` rArr log ( a + x ) = log y - log ( 1 - ay ) + log c `
` rArr log ( x + a ) ( 1- ay ) = log cy `
` rArr ( x + a ) ( 1- ay ) = cy `