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.
A solution to a system of equations is a value of x and a value of y that, when substituted into the equations, satisfies both equations at the same time. For the example above and is a solution to the system. This is easy enough to check. So, sure enough that pair of numbers is a solution to the system.