 Solution

One scalar initial condition is needed in each case, because each equation is first order and each regular solution has one integration constant. The usual Picard-Lindelöf theorem applies locally at a regular initial point. For part (c), this requires $x_0\ne 0$ and $y_0\ne 0$; a condition at an excluded singular endpoint cannot specify a classical solution there. Arbitrary distant boundary data also need not be compatible with one common interval of existence.