Here is a homogeneous equation in which the total degree of both the numerator and the denominator of the right-hand side is 2. The two parts of the solution list give branches of the integral curves in the form .
This plots both branches together, showing the complete integral curves for several values of C.
If an initial condition is specified, DSolve picks the branch that passes through the initial point. The DSolve::bvnul message indicates that one branch of the general solution (the lower branch in the previous graph) did not give a solution satisfying the given initial condition y3.