QUESTION: (x+1) dy/dx= x (y^2+1)
Solve by variable separation method.
put all the y's on the left anf all the x's on the right.
dy/(y^2+1) = x.dx/(x+1)
now integrate both sides.
Int 1/(y^2+1) dy = Int x/(x+1) dx
LHS
Int 1/(y^2+1) dy = atan(y)
RHS
Int x/(x+1) dx = Int 1 - 1/(x+1) dx = x - ln(x+1) - lnA (lnA = const of integration)
Int x/(x+1) dx = x - ln(A(x+1))
Equating both sides,
atan(y) = x - ln(A(x+1))
y = tan{x - ln(A(x+1))}