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.
Solve the differential equation y' = 2 x with the condition that y (0) = 1, that is, y = 1 when x = 0. The condition y (0) = 1 is called an initial condition and a differential equation with an initial condition is called an initial-value problem. Solve the initial-value problem y' = 2 x and y (0) = 1.