The solution of (x d y d x =y( y- x+1) ) is

Step-by-step solution

( aligned & x d x d x =y( y- x+1) & d y d x = y x ( y x +1 ) (i) aligned ) ( gathered Let, f(x, y)= y x ( y x +1 ) f(k x, k y)= k y k x ( k y k x +1 ) gathered ) (f(k x, k y)= y x ( y x +1 )=f(x, y) ) ( ) Given differentiate Equation is homogeneous ( array l p u t, y=v x d y d x =v+x d v d x array ) From Eq. (i), ( aligned & v+x d v d x = v x x ( v x x +1 ) & v+x d v d x =v( v+1) & v+x d v d x =v v+v & 1 v v d v= 1 x d x aligned ) Integrating on both sides, ( 1 v 1 v d v= 1 x d x ) Put, ( v=t ) ( aligned 1 v d v &