After selecting a character, the player then selects a starting stage through an overworld map.
To say that you have found the general homogeneous solution means that this function solves the homogeneous equation for every choice of the constant $C_1$ and every solution of the homogeneous equation is of this form for some choice of $C_1$. You can actually have more than one particular solution to a DEQ.
The differential equation is homogeneous if the function f(x,y) is homogeneous, that is- Check that the functions . are homogeneous. In order to solve this type of equation we make use of a substitution (as we did in case of Bernoulli equations). Indeed, consider the substitution . If f(x,y) is homogeneous, then we have